Groups with infinite FC-center have the Schmidt property

TL;DR

This paper proves that countable groups with infinite FC-center possess the Schmidt property, meaning they admit specific measure-preserving actions with non-trivial central sequences, extending to groups with property (T).

Contribution

It establishes that all countable groups with infinite FC-center have the Schmidt property, including those with property (T), which was previously unknown.

Findings

Countable groups with infinite FC-center have the Schmidt property.

Inner amenable groups with property (T) also have the Schmidt property.

The result links group structural properties to dynamical actions with central sequences.

Abstract

We show that every countable group with infinite FC-center has the Schmidt property, i.e., admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As its consequence, every countable, inner amenable group with property (T) has the Schmidt property.

Equations

$$1\to \mathcal{U}\to {\mathcal{G}}_{\tilde{\sigma }}\to X⋊\Gamma \to 1$$

$$\tilde{\sigma }((\tau ,g),(g^{-1}\tau ,h))=(\tau ,\tau (\sigma (g,h)))$$

$$\mu (T^{\circ }{A}_n△A_n)\to 0$$

$$\mu (\{x\in {\mathcal{G}}^0\mid (T_n\circ S)x\ne (S\circ T_n)x\})\to 0$$

$$Z_x=\{f:{\mathcal{G}}^x\to Y\mid f(g{h}^{-1})=\alpha (h)f(g)\text{for each}g\in {\mathcal{G}}^x\text{and each}h\in {\mathcal{S}}_{s(g)}\}$$

$$(gf)(g^{\prime })=f(g^{-1}{g}^{\prime })$$

$${\int }_X\nu (\alpha (T_{n}x)B△B)d\mu (x)\to 0$$

$$C=\{(x,(y_i{)}_{i=1}^m)\in X\times {\prod }_{i=1}^{m}Y\mid x\in A\text{and}{y}_i\in B_i\text{for each}i\in \{1,\dots ,l\}\},$$

$$({\tilde{T}}_n^{\circ }f)({\psi }_i(y))$$

$$=\alpha (T_n({\bar{\psi }}_i(x)))f({\psi }_i(x)),$$

$$\zeta (C\cap ({\tilde{T}}_n^{\circ }{)}^{-1}C)\ge \zeta (C_1)={\int }_{A_1\cap (T_n^{\circ }{)}^{-1}A}\prod _{i=1}^l\nu (\alpha (T_n({\bar{\psi }}_i(x)){)}^{-1}{B}_i\cap B_i))d\mu (x).$$

$$\displaystyle\int_{A_{1}\cap(T_{n}^{\circ})^{-1}A}\prod_{i=1}^{l}\biggl{(}\nu(B_{i})-\frac{1}{2}\nu(\alpha(T_{n}(\bar{\psi}_{i}(x)))^{-1}B_{i}\bigtriangleup B_{i})\biggr{)}\,d\mu(x)$$

$$>\zeta (C)-\mu (A\setminus (A_1\cap (T_n^{\circ }{)}^{-1}A))-\epsilon /2$$

$$\ge \zeta (C)-(\mu (A\setminus A_1)+\mu (A\setminus (T_n^{\circ }{)}^{-1}A))-\epsilon /2>\zeta (C)-2\epsilon$$

$$Z_k=\{x\in D\mid S^{\circ }x,(S^{\circ }{)}^{2}x,\dots ,(S^{\circ }{)}^{k-1}x\in D,(S^{\circ }{)}^{k}x\notin D\}.$$

$$\mu (Z_1)=\mu (D\setminus (S^{\circ }{)}^{-1}D)=\mu (S^{\circ }D\setminus D)\le \mu (E\setminus D)<\epsilon \mu (E)$$

$$\mu (\{x\in Z\mid Tx\ne Sx\})\le \mu (Z_1)<\epsilon \mu (E).$$

$$\mu (\{x\in B\mid Tx\ne Sx\})<\epsilon \mu (E)$$

$$\mu (\{x\in B\mid Tx\ne Sx\})\le \sum _{p>1/\epsilon }\mu ((S^{\circ }{)}^{-1}{B}_p^{\prime })=\sum _{p>1/\epsilon }{p}^{-1}\mu (B_p)\le \epsilon \mu (B)\le \epsilon \mu (E).$$

$$\mu (\{x\in C\mid Tx\ne Sx\})\le \mu ((S^{\circ }{)}^{N-1}{C}_0)+\mu (C\setminus {\bigcup }_{n=0}^{N-1}(S^{\circ }{)}^n{C}_0)<2\epsilon \mu (E).$$

$$\mu (\{x\in E\mid Tx\ne Sx\})<4\epsilon \mu (E)+\mu (E\setminus D)<5\epsilon \mu (E).\text{qed}$$

$$T^k=(T_n{)}^{k_n}\circ \cdots \circ (T_1{)}^{k_1}.$$

$$\mu (X\setminus {\mathcal{Q}}_{n-1}{F}_n)<{\epsilon }_n,$$

$$\mu (E_n^l\setminus D_n^l)<{\epsilon }_n\mu (E_n^l)$$

$${\Phi }_l=\{0,1,\dots ,l_1-1\}\times \{0,1,\dots ,l_2-1\}\times \cdots \times \{0,1,\dots ,l_{n-1}-1\}.$$

$$\mu (\{x\in E_n^l\mid T_{n}x\ne S_{n}x\})<5{\epsilon }_n\mu (E_n^l).$$

$$T_n((T^k{)}^{\circ }x)=((T^k\circ T_n)x)(T^{k}x{)}^{-1}$$

$$S_n((T^k{)}^{\circ }x)=((T^k\circ S_n)x)(T^{k}x{)}^{-1}=((T^k\circ T_n)x)(T^{k}x{)}^{-1}=T_n((T^k{)}^{\circ }x),$$

$$(X\setminus {\mathcal{Q}}_{n-1}{F}_n)\cup \bigcup _{l\in L_n}{\mathcal{Q}}_{n-1}\{x\in E_n^l\mid x\notin D_n^l\text{or}{T}_{n}x\ne S_{n}x\}.$$

$${\epsilon }_n+\sum _{l=(l_1,\dots ,l_{n-1})\in L_n}(l_1+\cdots +l_{n-1})(\mu (E_n^l\setminus D_n^l)+\mu (\{x\in E_n^l\mid T_{n}x\ne S_{n}x\}))$$

$$<{\epsilon }_n+\sum _{l=(l_1,\dots ,l_{n-1})\in L_n}(l_1+\cdots +l_{n-1})({\epsilon }_n+5{\epsilon }_n)\mu (E_n^l)\le 7{\epsilon }_n,$$

$$A_n^p=\{x\in X\mid \text{x is a p-periodic point of Sn∘}\}\text{and}{A}_n^{p,h}=\{x\in A_n^p\mid (S_n{)}^{p}x=h\}.$$

Full text

The first author was supported by JSPS Grant-in-Aid for Scientific Research, 17K05268.

The second author was supported by NSF Grant DMS 1855825

1. Introduction

Let $G$ be a countable group. Throughout the paper, we equip each countable group with the discrete topology unless otherwise stated. We say that $G$ is inner amenable if there exists a sequence $(\xi_{n})$ of non-negative unit vectors in $\ell^{1}(G)$ such that for each $g\in G$, we have $\|\xi_{n}^{g}-\xi_{n}\|_{1}\to 0$ and $\xi_{n}(g)\to 0$, where the function $\xi_{n}^{g}$ on $G$ is defined by $\xi_{n}^{g}(h)=\xi_{n}(ghg^{-1})$ for $h\in G$. Inner amenability was introduced by Effros [Ef] as a necessary condition for the group von Neumann algebra of $G$ to have property Gamma when $G$ satisfies the ICC condition. Inner amenability also arises in the context of p.m.p. actions of $G$. For brevity, by a p.m.p. action of $G$ we mean a measure-preserving action of $G$ on a standard probability space, where “p.m.p.” stands for “probability-measure-preserving”. Let us say that a free ergodic p.m.p. action of $G$ is Schmidt if the associated orbit equivalence relation admits a non-trivial central sequence in its full group. We say that $G$ has the Schmidt property if $G$ has a free ergodic p.m.p. action which is Schmidt. While the Schmidt property of $G$ implies inner amenability of $G$ ([JS, p.113]), the converse remains an open problem which was first posed by Schmidt [Sc, Problem 4.6]. Recent advances have lead to the resolution of some related long-standing problems concerning the relationship between inner amenability of groups and various kinds of central sequences ([Ki1] and [V]).

If the functions $\xi_{n}$ witnessing the inner amenability of $G$ are further required to be $G$-conjugation invariant, i.e., they each satisfy $\xi_{n}^{g}=\xi_{n}$ for all $g\in G$, then an algebraic constraint is imposed on $G$. In fact, the existence of such a sequence $(\xi_{n})$ is equivalent to $G$ having infinite FC-center. The FC-center of $G$ is defined as the subgroup of elements $g\in G$ whose centralizer, denoted by $C_{G}(g)$, is of finite index in $G$. The FC-center of $G$ is a normal (in fact, characteristic) subgroup of $G$.

In studying the structure of inner amenable groups, the second author [TD] introduced the AC-center of $G$, which is defined as the subgroup of elements $g\in G$ for which the quotient group $G/\bigcap_{h\in G}hC_{G}(g)h^{-1}$ is amenable. The AC-center of $G$ is also a characteristic subgroup of $G$ and contains the FC-center of $G$. If $G$ has infinite AC-center, then $G$ is inner amenable; this follows from the fact that for each element $g$ in the AC-center of $G$, the conjugation action of $G$ on the conjugacy class of $g$ factors through an action of the amenable group $G/\bigcap_{h\in G}hC_{G}(g)h^{-1}$. If $G$ is linear, or more generally fulfills a certain chain condition on its subgroups, then inner amenability of $G$ is equivalent to $G$ having infinite AC-center; in this case, the AC-center plays a crucial role in describing the structure of $G$, and this resulting structure can in turn be used to deduce that $G$ has the Schmidt property ([TD, Theorems 14 and 15]). However, there are many groups with infinite AC-center or FC-center, but which do not satisfy the relevant chain condition, so that the results of [TD] do not apply to these groups. In this paper, we solve Schmidt’s problem for them affirmatively:

Theorem 1.1**.**

Every countable group with infinite AC-center has the Schmidt property.

In fact, the Schmidt property for groups with infinite AC-center but finite FC-center follows from the constructions in [TD] (see Subsection 3.1). Thus, most of the proof of Theorem 1.1 is devoted to the case of groups with infinite FC-center.

The following corollary is an immediate consequence of Theorem 1.1 because every inner amenable group with property (T) has infinite FC-center.

Corollary 1.2**.**

Every countable, inner amenable group with property (T) has the Schmidt property.

It is widely known that property (T) is useful for constructing interesting examples regarding the non-existence of non-trivial central sequences in various contexts (e.g., [DV], [Ki1], [KTD], [PV] and [V]). By contrast, Corollary 1.2 says that there exist no counterexamples to Schmidt’s question among groups with property (T).

As mentioned above, the proof of Theorem 1.1 is reduced to that for a countable group $G$ with infinite FC-center. We present two constructions of a free p.m.p. Schmidt action of $G$. The first construction, given throughout Sections 2–5, stems from analysing central sequences for translation groupoids associated with (not necessarily free) p.m.p. actions. This analysis is of independent interest and yields by-products (Theorems 1.3 and 1.5) which do not follow from the second construction. The second construction, given in Section 6, is by way of ultraproducts of p.m.p. actions. While the first construction splits into cases depending on structure of $G$, the second construction does not split into cases and is more direct than the first.

A summary of the first construction. Let us describe some of the ingredients and by-products of the first construction. The construction is divided into two cases, depending on whether the FC-center has finite or infinite center. Let $G$ be a countable group with infinite FC-center $R$. If $R$ has finite center $C$, then $G$ admits a (not necessarily free) profinite action $G\curvearrowright(X,\mu)$ such that the quotient group $R/C$, which is infinite by assumption, acts freely. This action of $R/C$ leads us to find a central sequence in the full group of the groupoid $G\ltimes(X,\mu)$, similar to a construction of Popa-Vaes [PV] for residually finite groups with infinite FC-center. We need a further task to conclude that $G$ has the Schmidt property since the action $G\curvearrowright(X,\mu)$ is not necessarily free. We will return to this point after discussing the other case.

In the other case, the FC-center of $G$ has infinite center. The following construction is carried out after choosing some infinite abelian normal subgroup $A$ of $G$ contained in the FC-center of $G$. The group $A$ is not necessarily the center of the FC-center of $G$. We set $\Gamma=G/A$ and fix a section of the quotient map from $G$ onto $\Gamma$. The 2-cocycle $\sigma\colon\Gamma\times\Gamma\to A$ is then associated. The heart of the construction is to introduce the groupoid extension

[TABLE]

defined as follows: For some appropriate compact abelian metrizable group $L$, let $X$ be the group of homomorphisms from $A$ into $L$ and let $\mu$ be the normalized Haar measure on $X$. The conjugation $\Gamma\curvearrowright A$ induces the p.m.p. action $\Gamma\curvearrowright(X,\mu)$. We set $\mathcal{U}=X\times L$ and regard it as the bundle over $X$ with fiber $L$. Let $X\rtimes\Gamma$ be the translation groupoid and let $(X\rtimes\Gamma)^{(2)}$ be the set of composable pairs of $X\rtimes\Gamma$. The 2-cocycle $\tilde{\sigma}\colon(X\rtimes\Gamma)^{(2)}\to\mathcal{U}$ is then defined by

[TABLE]

for $\tau\in X$ and $g,h\in\Gamma$ (see [J, Theorem 1.1] for a related construction). This 2-cocycle $\tilde{\sigma}$ associates the groupoid $\mathcal{G}_{\tilde{\sigma}}$ that fits into the above exact sequence. Let $G$ act on $X$ via the quotient map from $G$ onto $\Gamma$. We then have a natural homomorphism $\eta\colon X\rtimes G\to\mathcal{G}_{\tilde{\sigma}}$ such that $\eta(\tau,a)=(\tau,\tau(a))\in\mathcal{U}$ for each $\tau\in X$ and $a\in A$. A crucial point is that if we prepare a free p.m.p. action $\mathcal{G}_{\tilde{\sigma}}\curvearrowright(Z,\zeta)$, then we can let $X\rtimes G$ and thus $G$ act on $(Z,\zeta)$ via $\eta$, so that the action of $A$ factors through the action of $\mathcal{U}$, which is easily handled since $L$ is compact. Moreover we can describe the stabilizer of a point of $Z$ in $G$ in terms of $\ker\eta$, which is contained in $X\rtimes A$.

Compact groups and their p.m.p. actions are utilized in many constructions of Schmidt actions such as in [DV], [Ki2], [Ki3], [KTD], [PV] and [TD]. They are useful on the basis of the following simple fact: For each p.m.p. action $K\curvearrowright(X,\mu)$ of a continuous (rather than compact) group $K$, each sequence converging to the identity in $K$ also converges to the identity in the automorphism group of $(X,\mu)$ in the weak topology. This weak convergence is necessary for a sequence in the full group to be central and is also sufficient if the sequence asymptotically commutes with each element of the acting group $G$.

Turning back to the general setup, let $G$ be an arbitrary countable group with infinite FC-center. Independent of whether the FC-center of $G$ has finite or infinite center, the above construction yields a p.m.p. action $G\curvearrowright(W,\omega)$ and a central sequence $(T_{n})$ in the full group of the translation groupoid $G\ltimes(W,\omega)$. The sequence $(T_{n})$ is non-trivial in the sense that the automorphism of $W$ induced by $T_{n}$ is nowhere the identity. We cannot yet conclude that $G$ has the Schmidt property because the action $G\curvearrowright(W,\omega)$ is not necessarily free.

Let us now simplify the setup as follows: Let $G$ be a countable group with a normal subgroup $M$ and a p.m.p. action $G\curvearrowright(X,\mu)$ such that $M$ acts on $X$ trivially and the quotient group $G/M$ acts on $X$ freely. Suppose that the groupoid $\mathcal{G}\coloneqq G\ltimes(X,\mu)$ is Schmidt, i.e., admits a central sequence $(T_{n})$ in its full group such that the automorphism of $X$ induced by $T_{n}$ is nowhere the identity. Under several additional assumptions, we then construct a free p.m.p. Schmidt action of $G$ as follows: After replacing $(T_{n})$ by another central sequence appropriately, we obtain the product subgroupoid $M\times\mathcal{R}<\mathcal{G}$ such that $\mathcal{R}$ is the groupoid generated by all $T_{n}$ and is also principal and hyperfinite. Pick a free p.m.p. action $M\curvearrowright(Y,\nu)$, let $M\times\mathcal{R}$ act on $(Y,\nu)$ via the projection from $M\times\mathcal{R}$ onto $M$, and co-induce the action $\mathcal{G}\curvearrowright(Z,\zeta)$ from the action $M\times\mathcal{R}\curvearrowright(Y,\nu)$. Then we have the lift of $(T_{n})$ into the translation groupoid $\mathcal{G}\ltimes(Z,\zeta)$. This lifted sequence is shown to be central in the full group, by using that $T_{n}$ acts on $Y$ trivially (see Proposition 2.4 for treatment of this fact in a more general framework). Moreover we can naturally define the p.m.p. action $G\curvearrowright(Z,\zeta)$ such that the associated groupoid $G\ltimes(Z,\zeta)$ is identified with $\mathcal{G}\ltimes(Z,\zeta)$. The action $G\curvearrowright(Z,\zeta)$ is free since the action $M\curvearrowright(Y,\nu)$ is free. Thus we obtain a free p.m.p. Schmidt action of $G$. This construction is flexible enough to apply to the more general setup, and we are able to deduce the Schmidt property for all groups with infinite FC-center. It also yields the following by-products:

Theorem 1.3** (Corollary 2.16).**

Let $G$ be a countable group and $M$ a finite central subgroup of $G$. Let $G/M\curvearrowright(X,\mu)$ be a free ergodic p.m.p. action and let $G$ act on $(X,\mu)$ through the quotient map from $G$ onto $G/M$. Suppose that the translation groupoid $G\ltimes(X,\mu)$ is Schmidt. Then $G$ has the Schmidt property.

Remark 1.4*.*

Let $G$ be a countable group and $M$ a finite central subgroup of $G$. It remains unsolved whether the Schmidt property of $G/M$ implies the Schmidt property of $G$ ([KTD, Question 5.16]). If $G/M$ has infinite AC-center, then $G$ also has the same property as well and thus has the Schmidt property (see Proposition 3.3 (ii) and related Remark 2.18).

Theorem 1.3 might be used to answer this question affirmatively: if there exists a free ergodic p.m.p. action $G/M\curvearrowright(X,\mu)$ which is Schmidt, along with a non-trivial central sequence in the full group of $(G/M)\ltimes(X,\mu)$ which lifts to a central sequence in the full group of $G\ltimes(X,\mu)$, then we can apply Theorem 1.3 and conclude that $G$ has the Schmidt property. While this lifting problem of central sequences is unsolved in full generality, we note that it is solved affirmatively for stability sequences in [Ki4].

A sequence $(g_{n})$ of elements of a countable group $G$ is called central if for each $h\in G$, $g_{n}$ commutes with $h$ for all sufficiently large $n$.

Theorem 1.5** (Corollary 2.17).**

If a countable group $G$ admits a central sequence diverging to infinity, then $G$ has the Schmidt property.

Remark 1.6*.*

Let $G$ be a countable group which admits a central sequence diverging to infinity. If $G$ has trivial center, then the Schmidt property for $G$ can be proved immediately as follows ([Ke2, Proposition 9.5]): Let $G$ act on the set $G\setminus\{e\}$ by conjugation, which induces the p.m.p. action of $G$ on the product space $X\coloneqq\prod_{G\setminus\{e\}}[0,1]$ equipped with the product measure $\mu$ of the Lebesgue measure. Then a central sequence in $G$ gives rise to a central sequence in the full group of $G\ltimes(X,\mu)$, and the action $G\curvearrowright(X,\mu)$ is essentially free since $G$ has trivial center.

Let $G$ be a countable group with infinite FC-center. Then given a sequence $(g_{n})$ in its FC-center diverging to infinity, each centralizer $C_{G}(g_{n})$ is of finite index in $G$, although the index of $C_{G}(g_{n})$ in $G$ possibly grows to infinity. In a sense, the $g_{n}$ may become less and less central in $G$ as $n$ increases. In this case, the above Bernoulli-like action of $G$ via conjugation $G\curvearrowright G\setminus\{e\}$ is not suitable for establishing the Schmidt property, and another approach must be taken.

An organization of the paper. In Section 2, we fix notation and terminology for discrete p.m.p. groupoids and describe co-induction of p.m.p. actions of discrete p.m.p. groupoids, extending the co-induction construction for actions of countable groups. As an application, we deduce the Schmidt property for a countable group $G$ under the assumption that $G$ admits a (not necessarily free) p.m.p. action $G\curvearrowright(X,\mu)$ such that the translation groupoid $G\ltimes(X,\mu)$ is Schmidt, together with some additional assumptions. In Section 3, we collect elementary properties of groups with infinite AC-center and reduce the proof of Theorem 1.1 to that for groups with infinite FC-center. Sections 4 and 5 are devoted to the first proof that groups with infinite FC-center have the Schmidt property. The proof in these two sections is divided into several cases, depending on the existence and structure of an infinite abelian normal subgroup of $G$ contained in the FC-center of $G$. An outline of the proof is given in Subsection 3.2. In Subsection 3.3, we exhibit examples of groups $G$ corresponding to each of the cases considered in Sections 4 and 5.

In Section 6, for a countable group with infinite FC-center, we give the second construction of a free p.m.p. Schmidt action, by way of ultraproducts.

In Appendix A, given an arbitrary countable abelian group $A$, we present a countable group with property (T) whose center is isomorphic to $A$. Our construction relies on the construction of Cornulier [C] and property (T) of the group $\mathit{SL}_{3}(\mathbb{Z}[t])\ltimes\mathbb{Z}[t]^{3}$, where $\mathbb{Z}[t]$ is the polynomial ring over $\mathbb{Z}$ in one indeterminate $t$. This result is useful in constructing interesting examples of groups with infinite FC-center along with Examples 3.6 and 3.7, while not being necessary for proving Theorem 1.1.

Throughout the paper, unless otherwise mentioned, all relations among Borel sets and maps are understood to hold up to null sets. Let $\mathbb{N}$ denote the set of positive integers.

Acknowledgments. We thank the anonymous referee for his/her careful reading of the paper and helpful corrections and suggestions, especially for Remark 2.3 and Lemma 2.7.

2. Central sequences in translation groupoids

2.1. Groupoids

We fix notation and terminology. Let $\mathcal{G}$ be a groupoid. We denote by $\mathcal{G}^{0}$ the unit space of $\mathcal{G}$ and denote by $r,s\colon\mathcal{G}\to\mathcal{G}^{0}$ the range and source maps of $\mathcal{G}$, respectively. For $x\in\mathcal{G}^{0}$, we set $\mathcal{G}^{x}=r^{-1}(x)$ and $\mathcal{G}_{x}=s^{-1}(x)$. For a subset $A\subset\mathcal{G}^{0}$, we set $\mathcal{G}_{A}=r^{-1}(A)\cap s^{-1}(A)$. The set $\mathcal{G}_{A}$ is then a groupoid with unit space $A$, with respect to the product inherited from $\mathcal{G}$. A groupoid $\mathcal{G}$ is called Borel if $\mathcal{G}$ is a standard Borel space, $\mathcal{G}^{0}$ is a Borel subset of $\mathcal{G}$, and the following maps are all Borel: the range and source maps, the multiplication map $(\gamma,\delta)\mapsto\gamma\delta$ defined for $\gamma,\delta\in\mathcal{G}$ with $s(\gamma)=r(\delta)$, and the inverse map $\gamma\mapsto\gamma^{-1}$. If the range and source maps are countable-to-one further, then $\mathcal{G}$ is called discrete. We mean by a discrete p.m.p. groupoid a pair $(\mathcal{G},\mu)$ of a discrete Borel groupoid $\mathcal{G}$ and a Borel probability measure $\mu$ on $\mathcal{G}^{0}$ such that $\int_{\mathcal{G}^{0}}c^{r}_{x}\,d\mu(x)=\int_{\mathcal{G}^{0}}c^{s}_{x}\,d\mu(x)$, where $c^{r}_{x}$ and $c^{s}_{x}$ are the counting measures on $\mathcal{G}^{x}$ and $\mathcal{G}_{x}$, respectively. The space $\mathcal{G}$ is then equipped with this common measure $\int_{\mathcal{G}^{0}}c^{r}_{x}\,d\mu(x)=\int_{\mathcal{G}^{0}}c^{s}_{x}\,d\mu(x)$.

A discrete p.m.p. groupoid is called principal if the map $\gamma\mapsto(r(\gamma),s(\gamma))$ is injective. Let $\mathcal{R}$ be a p.m.p. countable Borel equivalence relation on a standard probability space $(X,\mu)$. Then the pair $(\mathcal{R},\mu)$ is naturally a principal discrete p.m.p. groupoid with unit space $\mathcal{R}^{0}=\{\,(x,x)\mid x\in X\,\}$, which are simply identified with $X$ itself when there is no cause for confusion. The range and source maps are given by $r(x,y)=x$ and $s(x,y)=y$, respectively, and the multiplication and inverse operations are given by $(x,y)(y,z)=(x,z)$ and $(x,y)^{-1}=(y,x)$, respectively. We mean by a discrete p.m.p. equivalence relation on a standard probability space $(X,\mu)$ a p.m.p. countable Borel equivalence relation on $(X,\mu)$ equipped with this structure of a discrete p.m.p. groupoid.

Let $(\mathcal{G},\mu)$ be a discrete p.m.p. groupoid. A Borel subset $A\subset\mathcal{G}^{0}$ is called $\mathcal{G}$-invariant if $r(\mathcal{G}_{x})\subset A$ for $\mu$-almost every $x\in A$. We say that $(\mathcal{G},\mu)$ is ergodic if each $\mathcal{G}$-invariant Borel subset $A$ of $\mathcal{G}^{0}$ is $\mu$-null or $\mu$-conull. A local section of $\mathcal{G}$ is a Borel map $\phi\colon\mathrm{dom}(\phi)\rightarrow\mathcal{G}$, where $\mathrm{dom}(\phi)$ is a Borel subset of $\mathcal{G}^{0}$, such that $\phi(x)\in\mathcal{G}_{x}$ for each $x\in\mathrm{dom}(\phi)$ and the associated map $\phi^{\circ}\colon\mathrm{dom}(\phi)\rightarrow\mathcal{G}^{0}$, given by $\phi^{\circ}=r\circ\phi$, is injective. Two local sections are identified if their domains and values agree up to a $\mu$-null set. For two local sections $\phi\colon A\rightarrow\mathcal{G}$, $\psi\colon B\rightarrow\mathcal{G}$, the composition of them is the local section $\psi\circ\phi\colon(\phi^{\circ})^{-1}(\phi^{\circ}(A)\cap B)\rightarrow\mathcal{G}$ defined by $(\psi\circ\phi)(x)=\psi(\phi^{\circ}(x))\phi(x)$. The inverse of a local section $\phi\colon A\rightarrow\mathcal{G}$ is the local section $\phi^{-1}\colon\phi^{\circ}(A)\rightarrow\mathcal{G}$ defined by $\phi^{-1}(x)=\phi((\phi^{\circ})^{-1}(x))^{-1}$.

We denote by $[\mathcal{G}]$ the group of all local sections $\phi$ of $\mathcal{G}$ with $\mathrm{dom}(\phi)=\mathcal{G}^{0}$, and call $[\mathcal{G}]$ the full group of $(\mathcal{G},\mu)$. If the measure $\mu$ should be specified, then we denote it by $[(\mathcal{G},\mu)]$. In fact the full group is a group such that the product and inverse operations are given by the composition and inverse, respectively. For $\phi\in[\mathcal{G}]$ and a positive integer $n$, let $\phi^{n}$ denote the $n$ times composition of $\phi$ with itself, and let $\phi^{-n}$ denote the inverse of $\phi^{n}$. Let $\phi^{0}$ denote the trivial element of $[\mathcal{G}]$, i.e., the identity map on $\mathcal{G}^{0}$. We draw attention to distinction between the trivial element $\phi^{0}$ of $[\mathcal{G}]$ and the associated map $\phi^{\circ}=r\circ\phi$.

To each action $G\curvearrowright X$ of a group $G$ on a set $X$, the translation groupoid $\mathcal{G}=G\ltimes X$ is associated as follows: The set of groupoid elements is defined as $\mathcal{G}=G\times X$ with unit space $\{e\}\times X$, which is identified with $X$ if there is no cause of confusion. The range and source maps $r,s\colon\mathcal{G}\rightarrow\mathcal{G}^{0}$ are given by $r(g,x)=gx$ and $s(g,x)=x$, respectively. The multiplication and inverse operations are given by $(g,hx)(h,x)=(gh,x)$ and $(g,x)^{-1}=(g^{-1},gx)$, respectively. Suppose that $G$ is a countable group and $X$ is a standard Borel space equipped with a Borel probability measure $\mu$. If the action $G\curvearrowright X$ is further Borel and preserves $\mu$, then the pair $(G\ltimes X,\mu)$ is a discrete p.m.p. groupoid and is denoted by $G\ltimes(X,\mu)$. It is also denoted by $G\ltimes X$ for brevity if $\mu$ is understood from the context. If the action $G\curvearrowright(X,\mu)$ is essentially free, i.e., the stabilizer of almost every point of $X$ is trivial, then the groupoid $G\ltimes(X,\mu)$ is isomorphic to the associated orbit equivalence relation $\{\,(gx,x)\mid g\in G,\,x\in X\,\}$ via the map $(g,x)\mapsto(gx,x)$.

For each action $G\curvearrowright X$, we similarly define the groupoid $X\rtimes G$ such that the set of groupoid elements is $X\times G$ and the range and source of $(x,g)\in X\times G$ are $x$ and $g^{-1}x$, respectively. Then $X\rtimes G$ is isomorphic to $G\ltimes X$ via the map $(x,g)\mapsto(g,g^{-1}x)$.

Let $p\colon G\times X\rightarrow G$ be the projection. Then each local section $\phi$ of the groupoid $G\ltimes X$ is completely determined by the composed map $p\circ\phi\colon\mathrm{dom}(\phi)\rightarrow G$. Thus we will abuse notation and identify $\phi$ with $p\circ\phi$ if there is no cause of confusion. The group $G$ embeds into $[G\ltimes X]$ via the map $g\mapsto\phi_{g}$, where $\phi_{g}\colon X\rightarrow G$ is the constant map with value $g$.

2.2. Central sequences

Let $(\mathcal{G},\mu)$ be a discrete p.m.p. groupoid. A sequence $(A_{n})$ of Borel subsets of the unit space $\mathcal{G}^{0}$ is called asymptotically invariant for $(\mathcal{G},\mu)$ if

[TABLE]

for every $T\in[\mathcal{G}]$. A sequence $(T_{n})$ in the full group $[\mathcal{G}]$ is called central in $[\mathcal{G}]$ if $T_{n}$ asymptotically commutes with every $S\in[\mathcal{G}]$, i.e.,

[TABLE]

for every $S\in[\mathcal{G}]$.

Remark 2.1*.*

Let $G$ be a countable subgroup of $[\mathcal{G}]$ and suppose that $G$ generates $\mathcal{G}$, i.e., the minimal subgroupoid of $\mathcal{G}$ containing $G$ in its full group is equal to $\mathcal{G}$. Then a sequence $(A_{n})$ of Borel subsets of $\mathcal{G}^{0}$ is asymptotically invariant for $(\mathcal{G},\mu)$ if $\mu(gA_{n}\bigtriangleup A_{n})\to 0$ for every $g\in G$ ([JS, p.93]). Moreover a sequence $(T_{n})$ in $[\mathcal{G}]$ is central if and only if $T_{n}$ asymptotically commutes with every $g\in G$ and $\mu(T_{n}^{\circ}A\bigtriangleup A)\to 0$ for every Borel subset $A\subset X$ ([JS, Remark 3.3] or [Ki4, Lemma 2.3]). While these assertions are verified only for translation groupoids $G\ltimes(X,\mu)$ in the cited papers, the same proof is available for the above generalization.

We say that a discrete p.m.p. groupoid $(\mathcal{G},\mu)$ is Schmidt if there exists a central sequence $(T_{n})$ in $[\mathcal{G}]$ such that $\mu(\{\,x\in X\mid T_{n}^{\circ}x\neq x\,\})\to 1$. We say that a p.m.p. action $G\curvearrowright(X,\mu)$ of a countable group $G$ is Schmidt if the groupoid $G\ltimes(X,\mu)$ is Schmidt. If a countable group $G$ admits a free ergodic p.m.p. action which is Schmidt, then we say that $G$ has the Schmidt property. (N.B. A countable group, being a discrete p.m.p. groupoid on a singleton, is never Schmidt.) The following lemma implies that the Schmidt property of $G$ follows once we find a free p.m.p. Schmidt action of $G$ which may not be ergodic. We refer to [H, Section 6] for the ergodic decomposition of discrete p.m.p. groupoids.

Lemma 2.2**.**

Let $(\mathcal{G},\mu)$ be a discrete p.m.p. groupoid with the ergodic decomposition map $\pi\colon(\mathcal{G}^{0},\mu)\to(Z,\zeta)$ and the disintegration $\mu=\int_{Z}\mu_{z}\,d\zeta(z)$. Suppose that $(\mathcal{G},\mu)$ is Schmidt and let $(T_{n})$ be a central sequence in $[(\mathcal{G},\mu)]$ such that $\mu(\{\,x\in X\mid T_{n}^{\circ}x\neq x\,\})\to 1$. Then there exists a subsequence $(T_{n_{i}})$ of $(T_{n})$ such that for $\zeta$-almost every $z\in Z$, $(T_{n_{i}})$ is a central sequence in $[(\mathcal{G},\mu_{z})]$ such that $\mu_{z}(\{\,x\in X\mid T_{n_{i}}^{\circ}x\neq x\,\})\to 1$. Thus for $\zeta$-almost every $z\in Z$, the ergodic component $(\mathcal{G},\mu_{z})$ is Schmidt.

Proof.

Let $\mathcal{B}$ be the sigma field of Borel subsets of $\mathcal{G}^{0}$. Let $\{A_{k}\}$ be a countable subfamily of $\mathcal{B}$ which generates $\mathcal{B}$. Then for every $z\in Z$, the family $\{A_{k}\}$ generates a dense subfield in $\mathcal{G}^{0}$ with respect to $\mu_{z}$. Since $(T_{n})$ is central in $[(\mathcal{G},\mu)]$, we have $\int_{Z}\mu_{z}(T_{n}^{\circ}A_{k}\bigtriangleup A_{k})\,d\zeta(z)=\mu(T_{n}^{\circ}A_{k}\bigtriangleup A_{k})\to 0$ for each $k$. Thus after passing to a subsequence of $(T_{n})$, for $\zeta$-almost every $z\in Z$, we have $\mu_{z}(T_{n}^{\circ}A_{k}\bigtriangleup A_{k})\to 0$ for each $k$.

Applying the Lusin-Novikov uniformization theorem ([Ke1, Theorem 18.10]), we obtain a countable collection $\{\phi_{l}\}$ of local sections of $\mathcal{G}$ such that $\bigcup_{l}\phi_{l}(\mathrm{dom}(\phi_{l}))=\mathcal{G}$. Similarly to the above, after passing to a subsequence of $(T_{n})$, for $\zeta$-almost every $z\in Z$, we have $\mu_{z}(\{\,x\in X\mid(\phi_{l}\circ T_{n})x=(T_{n}\circ\phi_{l})x\,\})\to 1$ for each $l$ and $\mu_{z}(\{\,x\in X\mid T_{n}^{\circ}x\neq x\,\})\to 1$. The first convergence together with the convergence obtained in the last paragraph implies that $(T_{n})$ is a central sequence in $[(\mathcal{G},\mu_{z})]$ for $\zeta$-almost every $z\in Z$. ∎

2.3. Co-induced actions

Co-induction is a canonical method to obtain a p.m.p. action of a countable group from a p.m.p. action of its subgroup. We generalize this for p.m.p. actions of discrete p.m.p. groupoids.

Remark 2.3*.*

Formally we mean by an action of a groupoid $\mathcal{G}$ an action of $\mathcal{G}$ on a space $Z$ fibered over $\mathcal{G}^{0}$ such that each $g\in\mathcal{G}$ gives rise to an isomorphism from the fiber at the source of $g$ onto the fiber at the range of $g$. Then we say that $\mathcal{G}$ acts on the fibered space $Z$. We often obtain such an action of $\mathcal{G}$ from a groupoid homomorphism $\alpha\colon\mathcal{G}\to\mathrm{Aut}(Y)$ for some space $Y$, as follows: Let $Z=\mathcal{G}^{0}\times Y$ and regard it to be fibered over $\mathcal{G}^{0}$ via the projection. Then $\mathcal{G}$ acts on $Z$ by $g(s(g),y)=(r(g),\alpha(g)y)$. For simplicity we will often abuse terminology of actions, and call this action on the fibered space $Z$ an action of $\mathcal{G}$ on the space $Y$ (which is not fibered over $\mathcal{G}^{0}$ though) unless there is cause of confusion.

Let $(\mathcal{G},\mu)$ be a discrete p.m.p. groupoid and set $X=\mathcal{G}^{0}$. Let $\mathcal{S}$ be a Borel subgroupoid of $\mathcal{G}$ and suppose that $\mathcal{S}$ admits the measure-preserving action on a standard probability space $(Y,\nu)$ arising from a Borel homomorphism $\alpha\colon\mathcal{S}\to\mathrm{Aut}(Y,\nu)$. From this action of $\mathcal{S}$, we co-induce a p.m.p. action $\mathcal{G}\curvearrowright(Z,\zeta)$ as follows: For each $x\in X$, we set

[TABLE]

and define $Z$ as the disjoint union $Z=\bigsqcup_{x\in X}Z_{x}$. The set $Z$ is fibered with respect to the projection $p\colon Z\to X$ sending each element of $Z_{x}$ to $x$. The groupoid $\mathcal{G}$ acts on $Z$ by

[TABLE]

for $g\in\mathcal{G}_{x}$, $g^{\prime}\in\mathcal{G}^{r(g)}$ and $f\in Z_{x}$ with $x\in X$.

A measure-space structure on $Z$ is defined as follows: We have the decomposition of the unit space, $X=\bigsqcup_{m\in\mathbb{N}\cup\{\infty\}}X_{m}$, into the $\mathcal{G}$-invariant Borel subsets $X_{m}$ such that the index of $\mathcal{S}_{X_{m}}$ in $\mathcal{G}_{X_{m}}$ is the constant $m$. First suppose that $X=X_{m}$ for some $m\in\mathbb{N}\cup\{\infty\}$. Let $\{\psi_{i}\}_{i=1}^{m}$ be a family of choice functions for the inclusion $\mathcal{S}<\mathcal{G}$, i.e., a family of Borel maps $\psi_{i}\colon X\to\mathcal{G}$ such that for each $x\in X$, we have $\psi_{i}(x)\in\mathcal{G}^{x}$ and the family $\{\psi_{i}(x)\}_{i=1}^{m}$ is a complete set of representatives of all the equivalence classes in $\mathcal{G}^{x}$, where the equivalence relation on $\mathcal{G}^{x}$ is associated to the inclusion $\mathcal{S}<\mathcal{G}$ as follows: two elements $g,h\in\mathcal{G}^{x}$ are equivalent if and only if $g^{-1}h\in\mathcal{S}$. Then $Z$ is identified with the product space $X\times\prod_{i=1}^{m}Y$ under the map sending each $f\in Z_{x}$ with $x\in X$ to $(x,(f(\psi_{i}(x)))_{i})$. The measure-space structure on $Z$ is induced by this identification, where the space $X\times\prod_{i=1}^{m}Y$ is equipped with the product measure $\mu\times\prod_{i=1}^{m}\nu$. The action of $\mathcal{G}$ on $Z$ is Borel and preserves the probability measure on $Z$.

If $X$ is not necessarily equal to $X_{m}$ for some $m\in\mathbb{N}$, then as already stated, we have the decomposition $X=\bigsqcup_{m\in\mathbb{N}\cup\{\infty\}}X_{m}$ into $\mathcal{G}$-invariant Borel subsets. The set $Z$ is decomposed into the $\mathcal{G}$-invariant subsets $p^{-1}(X_{m})$, on which the measure-space structure is given in the way in the previous paragraph. Then the measure-space structure is also induced on $Z$, so that each $p^{-1}(X_{m})$ is Borel and the projection $p\colon Z\to X$ is measure-preserving.

Let $\zeta$ be the induced probability measure on $Z$. We define a discrete p.m.p. groupoid $(\mathcal{G},\mu)\ltimes(Z,\zeta)=(\tilde{\mathcal{G}},\tilde{\mu})$ as follows: The set of groupoid elements is the fibered product $\tilde{\mathcal{G}}\coloneqq\mathcal{G}\times_{X}Z$ with respect to the source map $s\colon\mathcal{G}\to X$ and the projection $p\colon Z\to X$. The unit space is $\tilde{\mathcal{G}}^{0}\coloneqq Z$ with measure $\tilde{\mu}\coloneqq\zeta$. The range and source maps are given by $\tilde{r}(g,z)=gz$ and $\tilde{s}(g,z)=z$, respectively, with groupoid operations given by $(gh,z)=(g,hz)(h,z)$ and $(g,z)^{-1}=(g^{-1},gz)$. Each element $T\in[\mathcal{G}]$ lifts to the element $\tilde{T}\in[\tilde{\mathcal{G}}]$ defined by $\tilde{T}z=(Tx,z)$ for $z\in Z_{x}$ with $x\in X$.

Let us recall the following fact from the proof of [TD, Theorem 15] or [KTD, Example 8.8]: Let $G$ be a countable group, $C$ a central subgroup of $G$, and $C\curvearrowright(Y,\nu)$ a p.m.p. action. We define $G\curvearrowright(Z,\zeta)$ as the action co-induced from the action $C\curvearrowright(Y,\nu)$. Then each sequence of elements of $C$ that converges to the identity in $\mathrm{Aut}(Y,\nu)$ is central in the full group of the groupoid $G\ltimes(Z,\zeta)$. We generalize this fact to the following:

Proposition 2.4**.**

Let $(\mathcal{G},\mu)$ be a discrete p.m.p. groupoid and set $X=\mathcal{G}^{0}$. Let $\mathcal{S}$ be a Borel subgroupoid of $\mathcal{G}$, $(Y,\nu)$ a standard probability space, and $\alpha\colon\mathcal{S}\to\mathrm{Aut}(Y,\nu)$ a Borel homomorphism. Let $\mathcal{G}\curvearrowright(Z,\zeta)$ denote the action co-induced from the action $\mathcal{S}\curvearrowright(X\times Y,\mu\times\nu)$ via $\alpha$. Let $(T_{n})$ be a central sequence in $[\mathcal{G}]$ such that each $T_{n}$ belongs to $[\mathcal{S}]$ and for each Borel subset $B\subset Y$, we have

[TABLE]

as $n\to\infty$. Then the sequence $(\tilde{T}_{n})$ of the lifts of $T_{n}$ is central in the full group of the groupoid $(\mathcal{G},\mu)\ltimes(Z,\zeta)$ defined above.

Proof.

Since $(T_{n})$ is central in $[\mathcal{G}]$, by the definition of lifts, $\tilde{T}_{n}$ asymptotically commutes with the lift of each $S\in[\mathcal{G}]$, i.e., $\zeta(\{\,z\in Z\mid(\tilde{S}\circ\tilde{T}_{n})z\neq(\tilde{T}_{n}\circ\tilde{S})z\,\})\to 0$ for each $S\in[\mathcal{G}]$. Hence it suffices to show that for each Borel subset $C\subset Z$, we have $\zeta(\tilde{T}_{n}^{\circ}C\bigtriangleup C)\to 0$ (Remark 2.1). We may suppose that the index of $\mathcal{S}$ in $\mathcal{G}$ is the constant $m\in\mathbb{N}\cup\{\infty\}$. Let $\{\psi_{i}\}_{i=1}^{m}$ be a family of choice functions for the inclusion $\mathcal{S}<\mathcal{G}$ and identify $Z$ with the product space $X\times\prod_{i=1}^{m}Y$ as being before the proposition. Then it suffices to show that $\zeta(\tilde{T}_{n}^{\circ}C\bigtriangleup C)\to 0$ for each cylindrical subset

[TABLE]

where $A\subset X$ and $B_{1},\ldots,B_{l}\subset Y$ are Borel subsets and $l$ is a positive integer with $l\leq m$.

Let $\varepsilon>0$. We set $\bar{\psi}_{i}=s\circ\psi_{i}$ and set $\phi_{i}(x)=\psi_{i}(x)^{-1}$ for $x\in X$. Since $\phi_{i}$ is the union of local sections of $\mathcal{G}$, the assumption on the central sequence $(T_{n})$ implies that there exists an $N\in\mathbb{N}$ such that if $n\geq N$, then

1. (1)

$\mu(T_{n}^{\circ}A\bigtriangleup A)<\varepsilon$, 2. (2)

$\int_{X}\nu(\alpha(T_{n}(\bar{\psi}_{i}(x)))B_{i}\bigtriangleup B_{i})\,d\mu(x)<\varepsilon/l$ for each $i\in\{1,\ldots,l\}$, and 3. (3)

$\mu(A_{1})>\mu(A)-\varepsilon$,

where $A_{1}$ is defined as the set of all elements $x\in A$ such that $(\phi_{i}\circ T_{n})x=(T_{n}\circ\phi_{i})x$ for each $i\in\{1,\ldots,l\}$. Fix $n\in\mathbb{N}$ with $n\geq N$. We show that $\tilde{T}_{n}^{\circ}f\in C$ if $f$ belongs to the set $C_{1}$, which is slightly smaller than $C$, of all elements $(x,(y_{i})_{i=1}^{m})\in X\times\prod_{i=1}^{m}Y$ such that

• •

$x\in A_{1}\cap(T_{n}^{\circ})^{-1}A$ and

• •

$y_{i}\in\alpha(T_{n}(\bar{\psi}_{i}(x)))^{-1}B_{i}\cap B_{i}$ for each $i\in\{1,\ldots,l\}$.

We pick $f=(x,(y_{i})_{i=1}^{m})\in C_{1}$ and set $y=T_{n}^{\circ}x$. For each $i\in\{1,\ldots,l\}$, regarding $f$ as a map from $\mathcal{G}^{x}$ to $Y$ belonging to the set $Z_{x}$, we have

[TABLE]

where the second equation follows from $x\in A_{1}$ and $\phi_{i}^{\circ}(x)=\bar{\psi}_{i}(x)$. The right hand side belongs to $B_{i}$ because $f(\psi_{i}(x))=y_{i}\in\alpha(T_{n}(\bar{\psi}_{i}(x)))^{-1}B_{i}$. Moreover $\tilde{T}_{n}^{\circ}f\in Z_{y}$ and $y\in A$ because $x\in(T_{n}^{\circ})^{-1}A$. Therefore $\tilde{T}_{n}^{\circ}f\in C$. As a result, we obtain the inequality

[TABLE]

The left hand side of this inequality is equal to $\zeta(C)-\zeta(\tilde{T}_{n}^{\circ}C\bigtriangleup C)/2$, and the right hand side is equal to

[TABLE]

by (1)–(3), where to deduce the first inequality, we use the inequality $|\prod_{i=1}^{l}a_{i}-\prod_{i=1}^{l}b_{i}|\leq\sum_{i=1}^{l}|a_{i}-b_{i}|$ for $a_{i},b_{i}\in[0,1]$. Therefore $\zeta(\tilde{T}_{n}^{\circ}C\bigtriangleup C)<4\varepsilon$. ∎

2.4. Construction of a free action

Under the assumption that a countable group $G$ admits a p.m.p. Schmidt action, in Theorem 2.5, we present a sufficient condition for $G$ to admit a free p.m.p. Schmidt action. Another sufficient condition will be given in Theorem 2.14 in Subsection 2.6. We remark that the analogous problem for stability in place of the Schmidt property is solved in [Ki3, Theorem 1.4] with a much simpler method.

For $p\in\mathbb{N}$ and a Borel automorphism $T$ of a standard Borel space $X$, we call a point $x\in X$ a $p$-periodic point of $T$ if $T^{p}x=x$ and $T^{i}x\neq x$ for all $i\in\mathbb{N}$ less than $p$. If a point $x\in X$ is a $p$-periodic point of $T$ for some $p\in\mathbb{N}$, then $x$ is called a periodic point of $T$ and the number $p$ is called the period of $x$. For possible constraints on periods of $T_{n}^{\circ}$ for a central sequence $(T_{n})$ in the full group, we refer to [KTD, Proposition 8.7].

Theorem 2.5**.**

Let $G$ be a countable group, $G\curvearrowright(X,\mu)$ a p.m.p. action and $\pi\colon(X,\mu)\to(\Omega,\eta)$ a $G$-equivariant measure-preserving map into a standard probability space $(\Omega,\eta)$. Suppose that for $\mu$-almost every $x\in X$, the stabilizer of $x$ in $G$ depends only on $\pi(x)$ and we thus have a subgroup $M_{\omega}$ of $G$ indexed by $\eta$-almost every $\omega\in\Omega$ such that for $\mu$-almost every $x\in X$, the stabilizer of $x$ in $G$ is equal to $M_{\pi(x)}$. We set $(\mathcal{G},\mu)=G\ltimes(X,\mu)$.

Suppose that there exists a central sequence $(S_{n})$ in $[\mathcal{G}]$ such that

• •

for all $n$, $S_{n}^{\circ}$ preserves each fiber of $\pi$, i.e., we have $\pi(S_{n}^{\circ}x)=\pi(x)$ for $\mu$-almost every $x\in X$, and

• •

$\mu(\{\,x\in X\mid S_{n}^{\circ}x\neq x,\,S_{n}x\in C_{G}(M_{\pi(x)})\,\})\to 1$* as $n\to\infty$,*

where for a subgroup $M<G$, we denote by $C_{G}(M)$ the centralizer of $M$ in $G$. For $p\in\mathbb{N}$, let $A_{n}^{p}\subset X$ be the set of $p$-periodic points of $S_{n}^{\circ}$. Suppose further that for each $p\in\mathbb{N}$, we have $\mu(A_{n}^{p})\to 0$ as $n\to\infty$. Then $G$ has the Schmidt property.

The proof of this theorem will be given after proving Lemmas 2.6 and 2.7 below. For a discrete p.m.p. groupoid $(\mathcal{G},\mu)$ and an element $T\in[\mathcal{G}]$, we say that $T$ is periodic if for $\mu$-almost every $x\in\mathcal{G}^{0}$, there exists a $p\in\mathbb{N}$ such that $x$ is a $p$-periodic point of $T^{\circ}$ and $T^{p}x=e$. We should emphasize that $T$ is not necessarily periodic even if every point of $X$ is a periodic point of the induced automorphism $T^{\circ}$.

Lemma 2.6**.**

Let $G$ be a countable group, $G\curvearrowright(X,\mu)$ a p.m.p. action and $\pi\colon(X,\mu)\to(\Omega,\eta)$ a $G$-equivariant measure-preserving map satisfying the assumption in the first paragraph in Theorem 2.5. We set $(\mathcal{G},\mu)=G\ltimes(X,\mu)$.

Pick $\varepsilon>0$ and $S\in[\mathcal{G}]$ such that $S^{\circ}$ preserves each fiber of $\pi$. Let $D$ and $E$ be Borel subsets of $X$ with $D\subset E$, and suppose that the following three conditions hold:

1. (1)

If $x\in D$, then $S^{\circ}x\neq x$ and $Sx\in C_{G}(M_{\pi(x)})$, and if $x\in D$ is further a $p$-periodic point of $S^{\circ}$ for some $p\in\mathbb{N}$, then either $p>1/\varepsilon$ or $S^{p}x=e$. 2. (2)

The inequality $\mu(E\setminus D)<\varepsilon\mu(E)$ holds. 3. (3)

The inclusion $S^{\circ}D\subset E$ holds.

Then there exists an element $T\in[\mathcal{G}_{E}]$ such that

1. (4)

$T$* is periodic,* 2. (5)

$T^{\circ}$* preserves each fiber of $\pi$ and $Tx\in C_{G}(M_{\pi(x)})$ for each $x\in E$, and* 3. (6)

$\mu(\{\,x\in E\mid Tx\neq Sx\,\})<5\varepsilon\mu(E)$.

Proof.

For a positive integer $k$, we set

[TABLE]

The sets $Z_{k}$ are mutually disjoint and satisfy $S^{\circ}Z_{k+1}\subset Z_{k}$ and $Z_{1}=D\setminus(S^{\circ})^{-1}D$. Thus

[TABLE]

by conditions (2) and (3).

We define a local section $T$ of $\mathcal{G}$ on $Z_{k}$ for $k\geq 2$, on $S^{\circ}Z_{2}$, and on $Z_{1}\setminus S^{\circ}Z_{2}$, respectively, as follows: It is defined so that $T$ is periodic and equal to $S$ on a subset as large as possible. If $x\in Z_{k}$ and $k\geq 2$, then we set $Tx=Sx$. For almost every $x\in S^{\circ}Z_{2}$, there is a maximal integer $k\geq 2$ such that $x\in(S^{\circ})^{k-1}Z_{k}$, and we let $y\in Z_{k}$ be the point with $x=(S^{\circ})^{k-1}y$ and set $Tx=(S^{k-1}y)^{-1}$. On $Z_{1}\setminus S^{\circ}Z_{2}$, we set $Tx=e$ for each point $x$ of that set. We defined the local section $T$ on the union $Z\coloneqq\bigcup_{k=1}^{\infty}Z_{k}$ and have the inequality

[TABLE]

We set $D_{1}=D\setminus Z$, which is $S^{\circ}$-invariant. Let $B$ be the set of points of $D_{1}$ that are $p$-periodic points of $S^{\circ}$ for some $p\in\mathbb{N}$. Let $C$ be the complement of $B$ in $D_{1}$, i.e., the set of aperiodic points of $S^{\circ}$ in $D_{1}$. For an integer $p\geq 2$, let $B_{p}$ denote the set of $p$-periodic points of $S^{\circ}$ in $B$. Then each $B_{p}$ is $S^{\circ}$-invariant, and $B$ is the disjoint union of the sets $B_{p}$ with $p\geq 2$ since $S^{\circ}x\neq x$ for each $x\in D$ by condition (1).

We extend the domain of $T$ to the set $B$ as follows. If $p\leq 1/\varepsilon$, then for each $x\in B_{p}$, we have $S^{p}x=e$ by condition (1) and we thus set $T=S$ on $B_{p}$, so that $T$ is periodic on it. Otherwise, i.e., if $p>1/\varepsilon$, then pick a Borel fundamental domain $B_{p}^{\prime}\subset B_{p}$ of the periodic automorphism $S^{\circ}|_{B_{p}}$. We set $Tx=Sx$ for $x\in B_{p}\setminus(S^{\circ})^{-1}B_{p}^{\prime}$ and set $Tx=(S^{p-1}(S^{\circ}x))^{-1}$ for $x\in(S^{\circ})^{-1}B_{p}^{\prime}$. Then $T^{p}x=e$ for each $x\in B_{p}$, and we have

[TABLE]

because

[TABLE]

We next define $T$ on $C$, the set of aperiodic points of $S^{\circ}$ in $D_{1}$. Let $N$ be a positive integer with $1/N<\varepsilon\mu(E)$. By the Rokhlin lemma, we can find a Borel subset $C_{0}\subset C$ such that $C_{0},S^{\circ}C_{0},\ldots,(S^{\circ})^{N-1}C_{0}$ are mutually disjoint and $\mu(C\setminus\bigcup_{n=0}^{N-1}(S^{\circ})^{n}C_{0})<\varepsilon\mu(E)$. We define $T$ on $C$ as follows: For $x\in C_{0}$ and $n\in\{0,1,\ldots,N-2\}$, we set $T((S^{\circ})^{n}x)=S((S^{\circ})^{n}x)$ and $T((S^{\circ})^{N-1}x)=(S^{N-1}x)^{-1}$. If $x\in C\setminus\bigcup_{n=0}^{N-1}(S^{\circ})^{n}C_{0}$, then we set $Tx=e$. Then $T$ is periodic on $C$ in the sense that each $x\in C$ is a $p$-periodic point of $T^{\circ}$ for some $p\in\mathbb{N}$ and we then have $T^{p}x=e$. We also have

[TABLE]

Finally we define $T$ on $E\setminus D$ by $Tx=e$ for each $x\in E\setminus D$. By construction $T^{\circ}$ is an automorphism of each of $Z$, $B$, $C$ and $E\setminus D$ and hence of $E$. Thus we defined $T\in[\mathcal{G}_{E}]$, which is periodic. This is a desired one. Indeed for each $x\in E$, the element $Tx$ is either $e$ or the product of some values of $S$, which belongs to $C_{G}(M_{\pi(x)})$ by condition (1). Therefore $T$ fulfills condition (5). By inequalities (2.1)–(2.3) and condition (2), we have

[TABLE]

In order to state the next lemma, we prepare the following terminology. Let $(\mathcal{G},\mu)$ be a discrete p.m.p. groupoid. For $T,S\in[\mathcal{G}]$, we say that $T$ and $S$ commute if $T\circ S=S\circ T$. Let $T=(T_{1},\ldots,T_{n})$ be a finite sequence of elements of $[\mathcal{G}]$ such that $T_{i}$ and $T_{j}$ commute for all $i$ and $j$. For $k=(k_{1},\ldots,k_{n})\in\mathbb{N}^{n}$, we set

[TABLE]

For $l=(l_{1},\ldots,l_{n})\in\mathbb{N}^{n}$, we say that a point $x\in\mathcal{G}^{0}$ is $(l,T)$-periodic if the following two conditions hold:

• •

For every $k=(k_{1},\ldots,k_{n})\in\mathbb{N}^{n}$, we have $(T^{k})^{\circ}x=x$ if and only if $k_{i}\equiv 0$ modulo $l_{i}$ for all $i\in\{1,\ldots,n\}$.

• •

If this equivalent condition holds, then we have $T^{k}x=e$ further.

For a discrete p.m.p. equivalence relation $\mathcal{Q}$ on a standard probability space $(X,\mu)$, we mean by a Borel transversal of $\mathcal{Q}$ a Borel subset of $X$ which meets each equivalence class of $\mathcal{Q}$ at exactly one point.

Lemma 2.7**.**

With the notation and the assumption in Theorem 2.5, let $\mathcal{R}$ be the orbit equivalence relation associated with the action $G\curvearrowright(X,\mu)$. Then there exists a central sequence $(T_{n})_{n\in\mathbb{N}}$ in $[\mathcal{G}]$ satisfying the following four conditions:

1. (i)

We have $\mu(\{\,x\in X\mid T_{n}^{\circ}x\neq x\,\})\to 1$. 2. (ii)

For each $n$, $T_{n}^{\circ}$ preserves each fiber of $\pi$ and $T_{n}x\in C_{G}(M_{\pi(x)})$ for all $x\in X$. 3. (iii)

For each $m$ and $n$, $T_{m}$ and $T_{n}$ commute. 4. (iv)

Let $\mathcal{Q}_{n}$ be the subrelation of $\mathcal{R}$ generated by $T_{1}^{\circ},\ldots,T_{n}^{\circ}$. Then there exists a Borel transversal $E_{n+1}\subset X$ of $\mathcal{Q}_{n}$ and its Borel partition $E_{n+1}=\bigsqcup_{l\in\mathbb{N}^{n}}E_{n+1}^{l}$ such that for each $l=(l_{1},\ldots,l_{n})\in\mathbb{N}^{n}$,

• •

every point of $E_{n+1}^{l}$ is $(l,T)$-periodic, where $T=(T_{1},\ldots,T_{n})$,

• •

$T_{n+1}^{\circ}E_{n+1}^{l}=E_{n+1}^{l}$, and

• •

*if $n\geq 2$, then $E_{n+1}^{l}\subset E_{n}^{(l_{1},\ldots,l_{n-1})}$. *

In particular, for each $n$, if $\mathcal{E}_{n}$ denotes the subgroupoid of $\mathcal{G}$ generated by $T_{1},\ldots,T_{n}$ (i.e., the minimal subgroupoid of $\mathcal{G}$ containing $T_{1},\ldots,T_{n}$ in its full group), then $\mathcal{E}_{n}$ and $\mathcal{Q}_{n}$ are isomorphic under the quotient map from $\mathcal{G}$ onto $\mathcal{R}$.

Proof.

Fix a decreasing sequence $(\varepsilon_{n})_{n\in\mathbb{N}}$ of positive numbers converging to [math]. We inductively construct a sequence $(T_{n},E_{n+1})_{n\in\mathbb{N}}$ of pairs satisfying conditions (ii)–(iv) and the inequality $\mu(\{\,x\in X\mid T_{n}x\neq S_{n}x\,\})<7\varepsilon_{n}$ for all $n$. This inequality implies condition (i) and also implies that the sequence $(T_{n})_{n\in\mathbb{N}}$ is central in $[\mathcal{G}]$.

In Theorem 2.5, we assume that for each $p\in\mathbb{N}$, we have $\mu(A_{n}^{p})\to 0$ as $n\to\infty$, where $A_{n}^{p}$ is the set of $p$-periodic points of $S_{n}^{\circ}$. After replacing $S_{1}$ with $S_{n}$ for a large $n$, we may assume that $\mu(X\setminus D_{1})<\varepsilon_{1}$, where $D_{1}$ is defined as the set of points $x\in X$ such that $S_{1}^{\circ}x\neq x$, $S_{1}x\in C_{G}(M_{\pi(x)})$, and if $x$ is a $p$-periodic point of $S_{1}^{\circ}$ for some $p\in\mathbb{N}$, then $p>1/\varepsilon_{1}$. Letting $D=D_{1}$ and $E=X$, we apply Lemma 2.6. We then obtain a periodic $T_{1}\in[\mathcal{G}]$ such that $T_{1}^{\circ}$ preserves each fiber of $\pi$; we have $T_{1}x\in C_{G}(M_{\pi(x)})$ for almost every $x\in X$; and $\mu(\{\,x\in X\mid T_{1}x\neq S_{1}x\,\})<5\varepsilon_{1}<7\varepsilon_{1}$. Since $T_{1}$ is periodic, we can find a Borel fundamental domain $E_{2}\subset X$ for the automorphism $T_{1}^{\circ}$ of $X$ and its Borel partition $E_{2}=\bigsqcup_{l\in\mathbb{N}}E_{2}^{l}$ such that $\mathcal{Q}_{1}E_{2}^{l}$ is equal to the set of $l$-periodic points of $T_{1}^{\circ}$, where $\mathcal{Q}_{1}$ is the subrelation of $\mathcal{R}$ generated by $T_{1}^{\circ}$. The first step of the induction completes.

Assuming that we have constructed $T_{1},\ldots,T_{n-1}$ and $E_{2},\ldots,E_{n}$, we construct $T_{n}$ and $E_{n+1}$. By induction hypothesis, the equivalence relation $\mathcal{Q}_{n-1}$ generated by $T_{1}^{\circ},\ldots,T_{n-1}^{\circ}$ admits a Borel transversal $E_{n}\subset X$ and its Borel partition $E_{n}=\bigsqcup_{l\in\mathbb{N}^{n-1}}E_{n}^{l}$ such that for each $l\in(l_{1},\ldots,l_{n-1})\in\mathbb{N}^{n-1}$, every point of $E_{n}^{l}$ is $(l,T)$-periodic, where we set $T=(T_{1},\ldots,T_{n-1})$. We choose a finite subset $L_{n}\subset\mathbb{N}^{n-1}$ such that $\mu(E_{n}^{l})>0$ for all $l\in L_{n}$ and

[TABLE]

where we set $F_{n}=\bigsqcup_{l\in L_{n}}E_{n}^{l}$. After replacing $S_{n}$ with $S_{m}$ for a large $m$, we may assume that

[TABLE]

for each $l\in L_{n}$ if $D_{n}^{l}$ is defined as the set of points $x\in E_{n}^{l}$ such that

• •

$x\in E_{n}^{l}\cap((S_{n}^{\circ})^{-1}E_{n}^{l})$, $S_{n}^{\circ}x\neq x$ and $S_{n}x\in C_{G}(M_{\pi(x)})$,

• •

if $x$ is a $p$-periodic point of $S_{n}^{\circ}$ for some $p\in\mathbb{N}$, then $p>1/\varepsilon_{n}$, and

• •

$(S_{n}\circ T^{k})x=(T^{k}\circ S_{n})x$ for each $k=(k_{1},\ldots,k_{n-1})\in\Phi_{l}$,

where we set

[TABLE]

Letting $D=D_{n}^{l}$ and $E=E_{n}^{l}$, we apply Lemma 2.6 for each $l\in L_{n}$. Then there exists a periodic $T_{n}\in[\mathcal{G}_{F_{n}}]$ such that $T_{n}^{\circ}$ preserves each $E_{n}^{l}$ with $l\in L_{n}$; we have $T_{n}x\in C_{G}(M_{\pi(x)})$ for almost every $x\in F_{n}$; and for each $l\in L_{n}$, we have

[TABLE]

We extend the local section $T_{n}$ to the set $\mathcal{Q}_{n-1}F_{n}$ so that it commutes with $T_{1},\ldots,T_{n-1}$. That is, if $l\in(l_{1},\ldots,l_{n-1})\in L_{n}$ and $x\in E_{n}^{l}$, then we set

[TABLE]

for $k=(k_{1},\ldots,k_{n-1})\in\Phi_{l}$. We note that by condition (iv) for $T_{1},\ldots,T_{n-1}$, which is an induction hypothesis, each point of $\mathcal{Q}_{n-1}F_{n}$ is uniquely written as $(T^{k})^{\circ}x$ for some $k\in\Phi_{l}$ and $x\in E_{n}^{l}$ with $l\in L_{n}$. Finally we define $T_{n}$ on $X\setminus\mathcal{Q}_{n-1}F_{n}$ by $T_{n}x=e$ for each point $x$ in that set. Then the element $T_{n}\in[\mathcal{G}]$ satisfies conditions (ii) and (iii). By construction, $T_{n}^{\circ}$ preserves each $E_{n}^{l}$ with $l\in L_{n}$ and also preserves the other $E_{n}^{l}$ with $l\in\mathbb{N}^{n-1}\setminus L_{n}$ since $T_{n}^{\circ}$ is the identity on it.

Let $\mathcal{Q}_{n}$ be the subrelation of $\mathcal{R}$ generated by $T_{1}^{\circ},\ldots,T_{n}^{\circ}$. We find a Borel transversal $E_{n+1}\subset X$ of $\mathcal{Q}_{n}$ satisfying condition (iv). Since $T_{n}^{\circ}$ preserves each $E_{n}^{l}$ with $l\in\mathbb{N}^{n-1}$ and is periodic, we can choose a Borel fundamental domain $B_{n}^{l}$ for the automorphism $T_{n}^{\circ}$ of $E_{n}^{l}$ and its Borel partition $B_{n}^{l}=\bigsqcup_{m\in\mathbb{N}}E_{n}^{l,m}$ such that $E_{n}^{l,m}$ consists of $m$-periodic points of $T_{n}^{\circ}$. Pick $l=(l_{1},\ldots,l_{n-1})\in\mathbb{N}^{n-1}$ and $m\in\mathbb{N}$ and put $k=(l_{1},\ldots,l_{n-1},m)\in\mathbb{N}^{n}$. If $l\in L_{n}$, we set $E_{n+1}^{k}=E_{n}^{l,m}$. Otherwise we have $B_{n}^{l}=E_{n}^{l,1}$. We then set $E_{n+1}^{k}=E_{n}^{l}$ or $E_{n+1}^{k}=\emptyset$, depending on $m=1$ or $m\neq 1$, respectively, and set $E_{n+1}=\bigsqcup_{k\in\mathbb{N}^{n}}E_{n+1}^{k}$. This partition fulfills condition (iv), except for the equation involving $T_{n+1}$ still not defined.

Finally we estimate the measure $\mu(\{\,x\in X\mid T_{n}x\neq S_{n}x\,\})$. If $x\in D_{n}^{l}$ with $l\in L_{n}$ and $T_{n}x=S_{n}x$, then for each $k=(k_{1},\ldots,k_{n-1})\in\Phi_{l}$, we have

[TABLE]

where the first equation follows from $x\in D_{n}^{l}$, the second one follows from $T_{n}x=S_{n}x$, and the third one holds by the definition of $T_{n}$. Hence we have $T_{n}=S_{n}$ on the equivalence class of $x$ in $\mathcal{Q}_{n-1}$. The set $\{\,x\in X\mid T_{n}x\neq S_{n}x\,\}$ is thus contained in the union

[TABLE]

By inequalities (2.4), (2.5) and (2.6), the measure of this union is less than

[TABLE]

where the sum $\sum_{l}(l_{1}+\cdots+l_{n-1})\mu(E_{n}^{l})$ over $l=(l_{1},\ldots,l_{n-1})\in L_{n}$ is equal to $\mu(\mathcal{Q}_{n-1}F_{n})$ by condition (iv) and hence at most $1$. We thus have $\mu(\{\,x\in X\mid T_{n}x\neq S_{n}x\,\})<7\varepsilon_{n}$. The induction completes. ∎

Proof of Theorem 2.5.

By Lemma 2.7, we obtain a central sequence $(T_{n})$ in $[\mathcal{G}]$ satisfying conditions (i)–(iv) in the lemma. Let $\mathcal{E}$ and $\mathcal{Q}$ be the unions $\bigcup_{n}\mathcal{E}_{n}$ and $\bigcup_{n}\mathcal{Q}_{n}$, respectively, where we use the symbols $\mathcal{E}_{n}$, $\mathcal{Q}_{n}$ in the lemma. Then $\mathcal{Q}$ is a subrelation of $\mathcal{R}$, and by condition (iv), $\mathcal{E}$ is a subgroupoid of $\mathcal{G}$ isomorphic to $\mathcal{Q}$ via the quotient map from $\mathcal{G}$ onto $\mathcal{R}$. Let $\mathcal{M}$ be the isotropy subgroupoid of $\mathcal{G}$, which is the bundle $\bigsqcup_{x\in X}M_{\pi(x)}$ over $X$. Let $\mathcal{M}\times_{X}\mathcal{E}$ be the fibered product with respect to the range map of $\mathcal{E}$. Then $(\mathcal{M}\times_{X}\mathcal{E},\mu)$ is a discrete p.m.p. groupoid with unit space $X$. Indeed the range and source of $(m,(g,x))\in\mathcal{M}\times_{X}\mathcal{E}$ are defined to be $gx$ and $x$, respectively. The product operation in $\mathcal{M}\times_{X}\mathcal{E}$ is defined by $(m,(g,hx))(l,(h,x))=(ml,(gh,x))$ for $(g,hx),(h,x)\in\mathcal{E}$ and $m,l\in M_{\pi(x)}$, where we note that $\pi(ghx)=\pi(hx)=\pi(x)$ since all $T_{n}^{\circ}$ preserve each fiber of $\pi$. Let $\mathcal{M}\vee\mathcal{E}$ be the subgroupoid of $\mathcal{G}$ generated by $\mathcal{M}$ and $\mathcal{E}$. By condition (ii), if $(g,x)\in\mathcal{E}$, then $g$ commutes with each element of $M_{\pi(x)}$. Therefore the map from $\mathcal{M}\times_{X}\mathcal{E}$ to $\mathcal{M}\vee\mathcal{E}$ sending $(m,(g,x))$ to $(mg,x)$ is a homomorphism and thus an isomorphism.

Let $\bar{\mathcal{M}}$ be the subgroupoid of $G\ltimes(\Omega,\eta)$ that is the bundle $\bigsqcup_{\omega\in\Omega}M_{\omega}$. We obtain the homomorphism from $\mathcal{M}\vee\mathcal{E}$ onto $\bar{\mathcal{M}}$ as the composition of the isomorphism from $\mathcal{M}\vee\mathcal{E}$ onto $\mathcal{M}\times_{X}\mathcal{E}$, with the projection from $\mathcal{M}\times_{X}\mathcal{E}$ onto $\bar{\mathcal{M}}$. Pick a Borel homomorphism $\alpha_{0}\colon\bar{\mathcal{M}}\to\mathrm{Aut}(Y,\nu)$ with some standard probability space $(Y,\nu)$ such that the associated action of $\bar{\mathcal{M}}$ on $(Y,\nu)$ is essentially free, i.e., we have $\alpha_{0}(m)y\neq y$ for almost every $y\in Y$ and almost every $m\in\bar{\mathcal{M}}\setminus\bar{\mathcal{M}}^{0}$, where $\bar{\mathcal{M}}$ is equipped with the measure $\int_{\Omega}c_{\omega}\,d\eta(\omega)$ with $c_{\omega}$ the counting measure on $M_{\omega}$. Such $\alpha_{0}$ is obtained as follows: Pick a free p.m.p. action $G\curvearrowright(Y,\nu)$. Via the projection from $G\ltimes(\Omega,\eta)$ onto $G$, we obtain the homomorphism from $G\ltimes(\Omega,\eta)$ into $\mathrm{Aut}(Y,\nu)$. Let $\alpha_{0}$ be its restriction to $\bar{\mathcal{M}}$. Then the action $\alpha_{0}$ is essentially free. Let $\mathcal{M}\vee\mathcal{E}$ act on $(Y,\nu)$ via the homomorphism from $\mathcal{M}\vee\mathcal{E}$ onto $\bar{\mathcal{M}}$, and denote this action by $\alpha\colon\mathcal{M}\vee\mathcal{E}\to\mathrm{Aut}(Y,\nu)$.

We now apply Proposition 2.4 by letting $\mathcal{S}=\mathcal{M}\vee\mathcal{E}$. Note that the central sequence $(T_{n})$ satisfies the assumption in the proposition, that is, for each Borel subset $B\subset Y$, we have $\int_{X}\nu(\alpha(T_{n}x)B\bigtriangleup B)\,d\mu(x)\to 0$ as $n\to\infty$, because $\mathcal{E}$ acts on $Y$ trivially and thus $\alpha(T_{n}x)$ is the identity for every $x\in X$. By the proposition, the sequence $(\tilde{T}_{n})$ of the lift of $T_{n}$ is central in the full group of the groupoid $(\tilde{\mathcal{G}},\tilde{\mu})$, where we let $\mathcal{G}\curvearrowright(Z,\zeta)$ be the action co-induced from the action $\alpha\colon\mathcal{M}\vee\mathcal{E}\to\mathrm{Aut}(Y,\nu)$ and let $(\tilde{\mathcal{G}},\tilde{\mu})=(\mathcal{G},\mu)\ltimes(Z,\zeta)$ be the groupoid associated with this co-induced action, introduced right before the proposition. Recall that $\tilde{\mathcal{G}}$ is the fibered product $\mathcal{G}\times_{X}Z$ with respect to the source map $s\colon\mathcal{G}\to X$ and is a groupoid with unit space $Z$.

If we define an action of $G$ on $Z$ by $gz=(g,x)z$ for $g\in G$ and $z\in Z_{x}$ with $x\in X$, then this action preserves the measure $\zeta$ and $(\tilde{\mathcal{G}},\tilde{\mu})$ is identified with the translation groupoid $G\ltimes(Z,\zeta)$ via the map $((g,x),z)\mapsto(g,z)$ for $g\in G$ and $z\in Z_{x}$ with $x\in X$. The action $G\curvearrowright(Z,\zeta)$ is free because the action of $\bar{\mathcal{M}}$ on $(Y,\nu)$ is free. Therefore we obtained the free p.m.p. action $G\curvearrowright(Z,\zeta)$ such that the groupoid $G\ltimes(Z,\zeta)$ is Schmidt. By Lemma 2.2, $G$ admits a free ergodic p.m.p. action which is Schmidt. ∎

2.5. Central sequences and periodic points

In Theorem 2.5, we assumed the central sequence $(S_{n})$ to satisfy the property that for each $p\in\mathbb{N}$, the set of $p$-periodic points of the automorphism $S_{n}^{\circ}$ has measure approaching [math]. On the other hand, in Theorem 2.14 in the next subsection, we focus on a central sequence $(S_{n})$ without this property. This subsection deals with such a central sequence toward the proof of Theorem 2.14.

In the rest of this subsection, we fix the following notation: Let $G$ be a countable group and $M$ a normal subgroup of $G$. Let $G/M\curvearrowright(X,\mu)$ be a free ergodic p.m.p. action and let $G$ act on $(X,\mu)$ through the quotient map from $G$ onto $G/M$. We set $(\mathcal{G},\mu)=G\ltimes(X,\mu)$.

Lemma 2.8**.**

Let $(S_{n})_{n\in\mathbb{N}}$ be a central sequence in $[\mathcal{G}]$. For $n,p\in\mathbb{N}$ and $h\in M$, we set

[TABLE]

Then

1. (i)

the sequence $(A_{n}^{p})_{n}$ is asymptotically invariant for $\mathcal{G}$. 2. (ii)

If $h$ is central in $G$, then the sequence $(A_{n}^{p,h})_{n}$ is asymptotically invariant for $\mathcal{G}$.

Proof.

Pick $\phi\in[\mathcal{G}]$. If $n$ is large, then the set

[TABLE]

has measure close to $1$. If $x\in A_{n}^{p}$ belongs to this set, then $(S_{n}^{\circ})^{i}(\phi^{\circ}x)=\phi^{\circ}((S_{n}^{\circ})^{i}x)$ for each $i\in\{1,\ldots,p\}$. The right hand side of this equation is not equal to $\phi^{\circ}x$ if $i<p$, and is equal to $\phi^{\circ}x$ if $i=p$. Hence $\phi^{\circ}x$ is a $p$-periodic point of $S_{n}^{\circ}$ and belongs to $A_{n}^{p}$. We thus have $\mu(\phi^{\circ}A_{n}^{p}\bigtriangleup A_{n}^{p})\to 0$ as $n\to\infty$. Assertion (i) follows.

To prove assertion (ii), we pick $g\in G$. If $n$ is large, then the set

[TABLE]

has measure close to $1$. If a point $x\in A_{n}^{p,h}$ belongs to this set, then

[TABLE]

and thus $(S_{n})^{p}(gx)=ghg^{-1}=h$ if $h$ is central in $G$. Combining this with assertion (i), we have $\mu(gA_{n}^{p,h}\bigtriangleup A_{n}^{p,h})\to 0$ as $n\to\infty$. Assertion (ii) follows. ∎

Lemma 2.9**.**

Let $(S_{n})_{n\in\mathbb{N}}$ be a central sequence in $[\mathcal{G}]$ and let $N$ be a normal subgroup of $G$. Then the sequence $(A_{n})$ defined by $A_{n}=\{\,x\in X\mid S_{n}x\in N\,\}$ is asymptotically invariant for $\mathcal{G}$.

Proof.

Pick $g\in G$. If $n$ is large, then for every point $x\in X$ outside a set of small measure, we have $(\phi_{g}\circ S_{n})x=(S_{n}\circ\phi_{g})x$, that is, $g(S_{n}x)=(S_{n}(gx))g$. Therefore if $x\in A_{n}$ further, then $S_{n}(gx)$ belongs to $gNg^{-1}=N$ and thus $gx\in A_{n}$. ∎

Remark 2.10*.*

Lemma 2.9 will be used in the proof of Lemma 2.11, by letting $N$ be the centralizer $C_{G}(M)$ of $M$ in $G$.

Let $(S_{n})_{n\in\mathbb{N}}$ be a central sequence in $[\mathcal{G}]$ and set $A_{n}=\{\,x\in X\mid S_{n}x\in C_{G}(M)\,\}$. While $(A_{n})$ is asymptotically invariant for $\mathcal{G}$ by Lemma 2.9, we further have $\mu(A_{n})\to 1$ if $M$ is finitely generated. Indeed if $F$ is a finite generating set of $M$ and $n$ is large enough, then for all $x\in X$ outside a set of small measure, we have $(\phi_{g}\circ S_{n})x=(S_{n}\circ\phi_{g})x$ for all $g\in F$ and hence $g(S_{n}x)=(S_{n}x)g$ since $M$ acts on $X$ trivially. Thus $S_{n}x$ commutes with every element of $M$.

Lemma 2.11**.**

Let $(S_{n})_{n\in\mathbb{N}}$ be a central sequence in $[\mathcal{G}]$ and $p\geq 2$ an integer. Let $h\in M$ and suppose that $h$ is central in $G$. We define $A_{n}\subset X$ as the set of $p$-periodic points $x$ of $S_{n}^{\circ}$ such that $(S_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(S_{n})^{p}x=h$. Suppose that $\mu(A_{n})$ is uniformly positive.

Then there exists a central sequence $(R_{n})$ in $[\mathcal{G}]$ such that if we define $B_{n}\subset X$ as the set of $p$-periodic points $x$ of $R_{n}^{\circ}$ such that $(R_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(R_{n})^{p}x=h$, then $\mu(B_{n})\to 1$.

Proof.

We follow the proof of [KTD, Lemma 5.3], patching the restrictions $S_{n}|_{A_{n}}$ together to obtain a desired $R\in[\mathcal{G}]$ after passing to an appropriate subsequence of $(S_{n})$.

Note that the equation $S_{n}^{\circ}A_{n}=A_{n}$ holds. Indeed let $x\in A_{n}$ and put $y=S_{n}^{\circ}x$. Then $y$ is a $p$-periodic point of $S_{n}^{\circ}$. The condition that $(S_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(S_{n})^{p}x=h\in C_{G}(M)$ implies that the value of $S_{n}$ at each point of the orbit of $x$ under iterations of $S_{n}^{\circ}$ belongs to $C_{G}(M)$. Thus $(S_{n})^{i}y\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$. We also have $((S_{n})^{p}y)(S_{n}x)=(S_{n})^{p+1}x=(S_{n}x)h=h(S_{n}x)$ and thus $(S_{n})^{p}y=h$. Therefore $y\in A_{n}$ and $S_{n}^{\circ}A_{n}\subset A_{n}$. The converse inclusion follows from this because $S_{n}^{\circ}$ is measure-preserving or we have $(S_{n}^{\circ})^{-1}=(S_{n}^{\circ})^{p-1}$ on $A_{n}$.

Since $A_{n}$ is asymptotically invariant for $\mathcal{G}$ by Lemmas 2.8 and 2.9, the sequence $(S_{n}^{\prime})$ in $[\mathcal{G}]$, defined by $S_{n}^{\prime}=S_{n}$ on $A_{n}$ and $S_{n}^{\prime}x=e$ for all $x\in X\setminus A_{n}$, is central in $[\mathcal{G}]$. After replacing $S_{n}$ with $S_{n}^{\prime}$, we may assume that $S_{n}x=e$ for all $x\in X\setminus A_{n}$. Then $(S_{n}^{\circ})^{p}$ is the identity on $X$. It suffices to show that for every $\varepsilon>0$ and every finite subset $F\subset[\mathcal{G}]$, there exists an $R\in[\mathcal{G}]$ such that $\mu(\{g\circ R\neq R\circ g\})<\varepsilon$ and $\mu(B)>1-\varepsilon$, where for $u,v\in[\mathcal{G}]$, we let $\{u\circ v\neq v\circ u\}$ be the set of points of $X$ on which $u\circ v$ and $v\circ u$ are not equal, and we define $B\subset X$ as the set of $p$-periodic points of $R^{\circ}$ such that $R^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $R^{p}x=h$.

Passing to a subsequence of $(S_{n})$, we may assume that the following conditions hold:

1. (1)

$\sum_{n}\mu(g^{\circ}A_{n}\bigtriangleup A_{n})<\varepsilon$ for all $g\in F$. 2. (2)

$\sum_{n}\mu(\{g\circ S_{n}\neq S_{n}\circ g\})<\varepsilon$ for all $g\in F$. 3. (3)

$\sum_{n}\sum_{k<n}\sum_{i=1}^{p-1}\mu((S_{n}^{\circ})^{i}A_{k}\bigtriangleup A_{k})<\varepsilon$.

Inequality (1) holds since the sequence $(A_{n})$ is asymptotically invariant for $\mathcal{G}$. The other two inequalities hold since the sequence $(S_{n})$ is central in $[\mathcal{G}]$. We set $C_{n}=\bigcup_{k<n}A_{k}$ and also set

[TABLE]

Note that the last union is disjoint. For each $n$, we have $S_{n}^{\circ}Y_{n}=Y_{n}$ because $(S_{n}^{\circ})^{p}$ is the identity on $X$ and $S_{n}^{\circ}A_{n}=A_{n}$. Then $Y_{n}\subset A_{n}\setminus C_{n}$ and $\sum_{n}\sum_{i=1}^{p-1}\mu((S_{n}^{\circ})^{i}C_{n}\bigtriangleup C_{n})<\varepsilon$ by inequality (3). Thus $\sum_{n}\mu((A_{n}\setminus C_{n})\setminus Y_{n})<\varepsilon$ and $\mu(\bigcup_{n}(A_{n}\setminus C_{n})\setminus Y)<\varepsilon$. By the definition of $C_{n}$, we have $\bigcup_{n}(A_{n}\setminus C_{n})=\bigcup_{n}A_{n}$, and this is equal to $X$ by [KTD, Lemma 5.1], where we use the assumption that $\mu(A_{n})$ is uniformly positive. Thus

1. (4)

$\mu(X\setminus Y)<\varepsilon$.

We pick $g\in F$ and estimate $\sum_{n}\mu(g^{\circ}Y_{n}\bigtriangleup Y_{n})$. Pick $y\in Y_{n}\setminus g^{\circ}Y_{n}$. Since $(g^{\circ})^{-1}y\not\in Y_{n}$, either $(g^{\circ})^{-1}y\not\in A_{n}$ or $(g^{\circ})^{-1}y\in D_{n}$, where we set $D_{n}=\bigcup_{i=0}^{p-1}(S_{n}^{\circ})^{i}C_{n}$. In the former case, we have $y\in A_{n}\setminus g^{\circ}A_{n}$. In the latter case, we have

[TABLE]

Let $N$ be a positive integer. We have

[TABLE]

By inequality (1), in the right hand side, the first term is less than $\varepsilon$. In general, for all Borel subsets $A,A^{\prime},B,B^{\prime}\subset X$, we have

[TABLE]

([KTD, Lemma 5.2]). This implies that the second term is less than or equal to

[TABLE]

where the first inequality follows from inequality (3) and the last inequality follows from inequality (1). Then $\sum_{n=1}^{N}\mu(Y_{n}\setminus g^{\circ}Y_{n})<(p+4)\varepsilon$ and therefore

1. (5)

$\sum_{n}\mu(Y_{n}\setminus g^{\circ}Y_{n})<(p+4)\varepsilon$ for all $g\in F$.

We define a map $R\colon X\to G$, patching the restrictions $S_{n}|_{Y_{n}}$ together as follows: For each $n$, we set $R=S_{n}$ on $Y_{n}$ and set $Rx=e$ if $x\in X\setminus Y$. Since $S_{n}^{\circ}$ preserves $Y_{n}$, the map $R^{\circ}$ is an automorphism of $X$ and hence $R$ is an element of $[\mathcal{G}]$. Let $B\subset X$ be the set of $p$-periodic points of $R^{\circ}$ such that $R^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $R^{p}x=h$. Since $S_{n}^{\circ}$ preserves $Y_{n}$ again and $Y_{n}$ is a subset of $A_{n}$, each point of $Y_{n}$ belongs to $B$ and therefore $Y=B$ and $\mu(B)>1-\varepsilon$ by inequality (4).

We pick $g\in F$ to estimate $\mu(\{g\circ R\neq R\circ g\})$. We have the following three inclusions:

[TABLE]

[TABLE]

[TABLE]

It follows from inequalities (2), (5) and (4) that

[TABLE]

The desired estimate is obtained after scaling $\varepsilon$. ∎

The following lemma is similar in appearance to the last lemma. The difference between them is the assumption on $\mu(A_{n})$ and the second condition in the definition of the set $B_{n}$. The following lemma deduces a stronger conclusion from the conclusion of the last lemma.

Lemma 2.12**.**

Let $(S_{n})_{n\in\mathbb{N}}$ be a central sequence in $[\mathcal{G}]$ and $p\geq 2$ an integer. Let $h\in M$ and suppose that $h$ is central in $G$. We define $A_{n}\subset X$ as the set of $p$-periodic points $x$ of $S_{n}^{\circ}$ such that $(S_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(S_{n})^{p}x=h$. Suppose that $\mu(A_{n})\to 1$.

Then there exists a central sequence $(R_{n})$ in $[\mathcal{G}]$ such that if we define $B_{n}\subset X$ as the set of $p$-periodic points $x$ of $R_{n}^{\circ}$ such that $(R_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(R_{n})^{p}x=e$, then $\mu(B_{n})\to 1$.

Proof.

We show that for all large $n\in\mathbb{N}$, if we choose a sufficiently large integer $m>n$ and set $R_{n}=(S_{m})^{-1}\circ S_{n}$, then the obtained sequence $(R_{n})$ works. Let $\varepsilon>0$ and fix a large $n\in\mathbb{N}$ such that $\mu(A_{n})>1-\varepsilon$. If $m$ is large enough, then $\mu(A_{m})>1-\varepsilon$ and $\mu(C)>1-\varepsilon$, where $C$ is the set of points $x\in X$ such that

• •

$(S_{n}\circ(S_{m})^{-1})x=((S_{m})^{-1}\circ S_{n})x$, and

• •

$((S_{m})^{-i}\circ(S_{n})^{i})x=((S_{m})^{-1}\circ S_{n})^{i}x$ for all $i\in\{1,\ldots,p\}$.

By [KTD, Lemma 5.6], for all $i\in\{1,\ldots,p-1\}$, we have

[TABLE]

as $m\to\infty$. Therefore for all $i\in\{1,\ldots,p-1\}$, since $(S_{n}^{\circ})^{i}x\neq x$ for all $x\in A_{n}$, after replacing $m$ with a larger integer, we may assume that there exists a Borel subset $A_{n}^{\prime}\subset A_{n}$ such that $\mu(A_{n}\setminus A_{n}^{\prime})<\varepsilon$ and $(S_{m}^{\circ})^{i}x\neq(S_{n}^{\circ})^{i}x$ for all $x\in A_{n}^{\prime}$. We set

[TABLE]

Then $\mu(D)>1-(3+p)\varepsilon$. We set $R=(S_{m})^{-1}\circ S_{n}$ and define $B\subset X$ as the set of $p$-periodic points of $R^{\circ}$ such that $R^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $R^{p}x=e$. We claim that $D\subset B$. This completes the proof of the lemma. Pick $x\in D$. We first show that $x$ is a $p$-periodic point of $R^{\circ}$ and $R^{p}x=e$. For each $i\in\{1,\ldots,p-1\}$, it follows from $x\in A_{n}^{\prime}$ that $(S_{m}^{\circ})^{i}x\neq(S_{n}^{\circ})^{i}x$, and follows from $x\in C$ that

[TABLE]

Hence $(R^{\circ})^{i}x\neq x$. We also have

[TABLE]

where the second equation follows from $x\in C$, the third equation follows from $x\in A_{n}$, and the last equation follows from $x\in A_{m}=(S_{m}^{\circ})^{p}A_{m}$. Finally for each $i\in\{1,\ldots,p-1\}$, we have

[TABLE]

which belongs to $C_{G}(M)$ because $x\in A_{n}\cap(S_{n}^{\circ})^{-i}A_{m}$ and the set $A_{m}$ is preserved by $S_{m}^{\circ}$, as shown in the second paragraph of the proof of Lemma 2.11. ∎

Combining Lemmas 2.11 and 2.12, we obtain the following corollary, which also reminds us of the notation fixed in the beginning of this subsection.

Corollary 2.13**.**

Let $G$ be a countable group and $M$ a normal subgroup of $G$. Let $G/M\curvearrowright(X,\mu)$ be a free ergodic p.m.p. action and let $G$ act on $(X,\mu)$ through the quotient map from $G$ onto $G/M$. We set $(\mathcal{G},\mu)=G\ltimes(X,\mu)$. Let $(S_{n})$ be a central sequence in $[\mathcal{G}]$ and $p\geq 2$ an integer. Let $h\in M$ and suppose that $h$ is central in $G$. We define $A_{n}\subset X$ as the set of $p$-periodic points $x$ of $S_{n}^{\circ}$ such that $(S_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(S_{n})^{p}x=h$. Suppose that $\mu(A_{n})$ is uniformly positive.

Then there exists a central sequence $(R_{n})$ in $[\mathcal{G}]$ such that if we define $B_{n}\subset X$ as the set of $p$-periodic points $x$ of $R_{n}^{\circ}$ such that $(R_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(R_{n})^{p}x=e$, then $\mu(B_{n})\to 1$.

2.6. A variant construction

Continuing from Subsection 2.4, we present another sufficient condition for a countable group $G$ to admit a free p.m.p. Schmidt action, under the assumption that $G$ admits a p.m.p. Schmidt action. In the following theorem, we assume the given p.m.p. action $G\curvearrowright(X,\mu)$ to be ergodic, as opposed to Theorem 2.5. This is because the proof uses certain asymptotically invariant sequences of subsets, which are better controlled if the action is ergodic.

Theorem 2.14**.**

Let $G$ be a countable group and $M$ a normal subgroup of $G$. Let $G/M\curvearrowright(X,\mu)$ be a free ergodic p.m.p. action and let $G$ act on $(X,\mu)$ through the quotient map from $G$ onto $G/M$. We set $(\mathcal{G},\mu)=G\ltimes(X,\mu)$.

Let $(S_{n})$ be a central sequence in $[\mathcal{G}]$, let $p\geq 2$ be an integer, and let $L<M$ be a finite subgroup which is central in $G$. We define $A_{n}\subset X$ as the set of $p$-periodic points of $S_{n}^{\circ}$ such that $(S_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(S_{n})^{p}x\in L$. Suppose that $\mu(A_{n})$ is uniformly positive. Then $G$ has the Schmidt property.

The scheme of the proof of this theorem is the same as that for Theorem 2.5. Lemma 2.6 will be used in the following lemma, which is analogous to Lemma 2.7:

Lemma 2.15**.**

With the notation and the assumption in Theorem 2.14, let $\mathcal{R}$ be the orbit equivalence relation associated with the action $G/M\curvearrowright(X,\mu)$. Then there exist a central sequence $(T_{n})_{n\in\mathbb{N}}$ in $[\mathcal{G}]$ and a sequence $(E_{n+1})_{n\in\mathbb{N}}$ of Borel subsets of $X$ satisfying conditions (i), (iii) and (iv) in Lemma 2.7 together with the following condition:

1. $\textrm{(ii)}^{\prime}$

For each $n$ and each $x\in X$, we have $T_{n}x\in C_{G}(M)$.

Proof.

The desired sequence $(T_{n},E_{n+1})_{n\in\mathbb{N}}$ is constructed by induction, similarly to the proof of Lemma 2.7. Fix a decreasing sequence $(\varepsilon_{n})_{n\in\mathbb{N}}$ of positive numbers converging to [math]. We inductively construct a sequence $(T_{n},E_{n+1})_{n\in\mathbb{N}}$ satisfying conditions $\textrm{(ii)}^{\prime}$, (iii) and (iv) and satisfying the inequality $\mu(\{\,x\in X\mid T_{n}x\neq S_{n}x\,\})<7\varepsilon_{n}$ for all $n$. Let $p$ be the integer in Theorem 2.14. Since $L$ is finite, by Corollary 2.13, we may assume without loss of generality that $\mu(B_{n})\to 1$, where we define $B_{n}\subset X$ as the set of $p$-periodic points $x$ of $S_{n}^{\circ}$ such that $(S_{n})^{i}x\in C_{G}(M)$ for all $i\in\{1,\ldots,p-1\}$ and $(S_{n})^{p}x=e$.

To construct $T_{1}$, we set $D_{1}=B_{1}$. After replacing $S_{1}$ with $S_{n}$ for a large $n$, we may assume that $\mu(X\setminus D_{1})<\varepsilon_{1}$. We apply Lemma 2.6 by letting $D=D_{1}$ and $E=X$ and letting $\Omega$ be a singleton. Then we obtain a periodic $T_{1}\in[\mathcal{G}]$ such that $T_{1}x\in C_{G}(M)$ for almost every $x\in X$ and $\mu(\{\,x\in X\mid T_{1}x\neq S_{1}x\,\})<5\varepsilon_{1}<7\varepsilon_{1}$. Since $T_{1}$ is periodic, we can find a Borel fundamental domain $E_{2}\subset X$ for the automorphism $T_{1}^{\circ}$ of $X$ and its Borel partition $E_{2}=\bigsqcup_{l\in\mathbb{N}}E_{2}^{l}$ such that $\mathcal{Q}_{1}E_{2}^{l}$ is equal to the set of $l$-periodic points of $T_{1}^{\circ}$, where $\mathcal{Q}_{1}$ is the subrelation of $\mathcal{R}$ generated by $T_{1}^{\circ}$. The first step of the induction completes.

Assuming that we have constructed $T_{1},\ldots,T_{n-1}$ and $E_{2},\ldots,E_{n}$, we construct $T_{n}$ and $E_{n+1}$. Let $\mathcal{Q}_{n-1}$ be the subrelation of $\mathcal{R}$ generated by $T_{1}^{\circ},\ldots,T_{n-1}^{\circ}$. By induction hypothesis, we have a Borel transversal $E_{n}\subset X$ of $\mathcal{Q}_{n-1}$ and its Borel partition $E_{n}=\bigsqcup_{l\in\mathbb{N}^{n-1}}E_{n}^{l}$. We choose a finite subset $L_{n}\subset\mathbb{N}^{n-1}$ and set $F_{n}=\bigsqcup_{l\in L_{n}}E_{n}^{l}$ as in the proof of Lemma 2.7. After replacing $S_{n}$ with $S_{m}$ for a sufficiently large $m$, for each $l\in L_{n}$, we define $D_{n}^{l}$ as the set of points $x\in E_{n}^{l}\cap((S_{n}^{\circ})^{-1}E_{n}^{l})\cap B_{n}$ such that $(S_{n}\circ T^{k})x=(T^{k}\circ S_{n})x$ for each $k=(k_{1},\ldots,k_{n-1})\in\Phi_{l}$, where we set $T^{k}=(T_{n-1})^{k_{n-1}}\circ\cdots\circ(T_{2})^{k_{2}}\circ(T_{1})^{k_{1}}$ and define $\Phi_{l}$ as before. Letting $D=D_{n}^{l}$ and $E=E_{n}^{l}$ and letting $\Omega$ be a singleton, we apply Lemma 2.6 for each $l\in L_{n}$ and obtain a periodic $T_{n}\in[\mathcal{G}_{F_{n}}]$. The rest of the construction of $T_{n}\in[\mathcal{G}]$, whose domain is extended to $X$, and a Borel transversal $E_{n+1}$ of $\mathcal{Q}_{n}$ is a verbatim translation of that in the proof of Lemma 2.7. ∎

Proof of Theorem 2.14.

The proof is a verbatim translation of that of Theorem 2.5, where we apply Lemma 2.15 in place of Lemma 2.7 and let $\Omega$ be a singleton. We note that the groupoid $\mathcal{M}\times_{X}\mathcal{E}$ in that proof then reduces to the direct product $M\times\mathcal{E}$. ∎

We now prove Theorems 1.3 and 1.5 stated in Section 1.

Corollary 2.16**.**

Let $G$ be a countable group and $M$ a finite central subgroup of $G$. Let $G/M\curvearrowright(X,\mu)$ be a free ergodic p.m.p. action and let $G$ act on $(X,\mu)$ through the quotient map from $G$ onto $G/M$. If the action $G\curvearrowright(X,\mu)$ is Schmidt, then $G$ has the Schmidt property.

Proof.

By assumption, we have a central sequence $(S_{n})$ in $[G\ltimes(X,\mu)]$ such that $\mu(\{\,x\in X\mid S_{n}^{\circ}x\neq x\,\})\to 1$, We will apply Theorem 2.5 or 2.14. The most remarkable difference between the assumptions in those two theorems is the condition on the set $A_{n}^{p}$ of $p$-periodic points of $S_{n}^{\circ}$ and its measure. Passing to a subsequence of $(S_{n})$, we may assume that either $\mu(A_{n}^{p})\to 0$ for every integer $p\geq 2$, or there is some integer $p\geq 2$ for which the values $\mu(A_{n}^{p})$ are uniformly positive. If the former holds, then we apply Theorem 2.5 by letting $\Omega$ be a singleton. We note that $C_{G}(M)=G$ since $M$ is central in $G$. If the latter holds, then we apply Theorem 2.14 by letting $L=M$. Thus the corollary follows from the theorems. ∎

Recall that a sequence $(g_{n})$ in a countable group $G$ is called central if for each $h\in G$, $g_{n}$ commutes with $h$ for all sufficiently large $n$. The following is an immediate application of Corollary 2.16:

Corollary 2.17**.**

If a countable group $G$ admits a central sequence diverging to infinity, then $G$ has the Schmidt property.

Proof.

Let $G$ act on the set $G\setminus\{e\}$ by conjugation, which induces the p.m.p. action of $G$ on the product space $X\coloneqq\prod_{G\setminus\{e\}}[0,1]$ equipped with the product measure $\mu$ of the Lebesgue measure. We may assume that $G$ has finite center because otherwise the Schmidt property of $G$ is shown in [KTD, Example 8.8]. Let $C$ be the center of $G$. Then $C$ acts on $X$ trivially and the induced action $G/C\curvearrowright(X,\mu)$ is essentially free. By assumption, we have a central sequence $(g_{n})$ in $G$ diverging to infinity, and we may assume that none of $g_{n}$ belongs to $C$. Then by Remark 2.1, $(g_{n})$ is a central sequence in the full group $[G\ltimes(X,\mu)]$ such that $\mu(\{\,x\in X\mid g_{n}x\neq x\,\})=1$ for all $n$. Thus Corollary 2.16 is applied to $G$ and its finite center $C$. ∎

Remark 2.18*.*

Let $G$ be a countable group. If $M$ is a finite central subgroup of $G$ and the quotient group $G/M$ admits a central sequence diverging to infinity, then $G$ also admits such a sequence and thus has the Schmidt property by Corollary 2.17.

To show this, choose a section $\mathsf{s}\colon G/M\to G$ of the quotient map. Let $(g_{n})$ be a central sequence in $G/M$ diverging to infinity. For each $h\in G$, the commutator $[\mathsf{s}(g_{n}),h]$ belongs to $M$ if $n$ is large enough. Since $M$ is finite, after passing to a subsequence, we may assume that for each $h\in G$, the element $[\mathsf{s}(g_{n}),h]$ is independent of $n$. Then the sequence $(\mathsf{s}(g_{n})\mathsf{s}(g_{1})^{-1})$ is central in $G$ and diverges to infinity.

3. Groups with infinite AC-center

3.1. Reduction to the proof for groups with infinite FC-center

We collect basic properties of groups with infinite AC-center. For a subset $S$ of a group $G$, we denote by $C_{G}(S)$ the centralizer of $S$ in $G$ and denote by $\langle S\rangle_{G}$ the normal closure of $S$ in $G$, i.e., the minimal normal subgroup of $G$ containing $S$. If $S$ consists of elements $g_{1},\ldots,g_{n}$, then $C_{G}(S)$ and $\langle S\rangle_{G}$ are also denoted by $C_{G}(g_{1},\ldots,g_{n})$ and $\langle g_{1},\ldots,g_{n}\rangle_{G}$, respectively.

Lemma 3.1**.**

Let $G$ be a countable group and denote by $R$ the AC-center of $G$, i.e., the set of elements $g\in G$ such that the quotient group $G/C_{G}(\langle g\rangle_{G})$ is amenable. Then

1. (i)

the set $R$ is a normal subgroup of $G$. 2. (ii)

For each finite subset $S\subset R$, the quotient group $G/C_{G}(\langle S\rangle_{G})$ is amenable. 3. (iii)

The group $R$ is amenable. 4. (iv)

The group $R$ is generated by all normal subgroups $M$ of $G$ such that $G/C_{G}(M)$ is amenable. Therefore $R$ is equal to the AC-center introduced in **[TD, 0.G]**.

Proof.

Although some assertions in the lemma are proved in [TD, Theorem 13], we give a proof for the reader’s convenience. For the ease of symbols, in this proof, let us write $\bar{C}(g)$ and $\bar{C}(S)$ for $C_{G}(\langle g\rangle_{G})$ and $C_{G}(\langle S\rangle_{G})$, respectively, given $g\in G$ and $S\subset G$. By its definition the set $R$ contains the trivial element and is closed under inverse. If $r,s\in R$, then $\bar{C}(r)\cap\bar{C}(s)<\bar{C}(rs)$. Thus $G/(\bar{C}(r)\cap\bar{C}(s))$ surjects onto $G/\bar{C}(rs)$ and injects into $G/\bar{C}(r)\times G/\bar{C}(s)$ diagonally. The last group is amenable and thus $rs\in R$. Hence $R$ is a subgroup of $G$, and by its definition $R$ is normal in $G$. Assertion (i) follows.

If $S$ consists of finitely many elements $r_{1},\ldots,r_{n}\in R$, then $G/\bar{C}(S)$ diagonally injects into the direct product $G/\bar{C}(r_{1})\times\cdots\times G/\bar{C}(r_{n})$, which is amenable. Thus $G/\bar{C}(S)$ is amenable, and assertion (ii) follows. Moreover the group $\langle S\rangle$ generated by $S$ admits the homomorphism into $G/\bar{C}(S)$ induced by the inclusion into $G$, whose kernel is $\langle S\rangle\cap\bar{C}(S)$ and thus abelian. Hence $\langle S\rangle$ is amenable, and assertion (iii) follows.

Let $\mathcal{M}$ be the set of normal subgroups $M$ of $G$ such that $G/C_{G}(M)$ is amenable, and let $R_{1}$ be the group generated by all members of $\mathcal{M}$. If $r\in R$, then $\langle r\rangle_{G}\in\mathcal{M}$ and thus $r\in R_{1}$. To show the converse, we note that if $M_{1},M_{2}\in\mathcal{M}$, then the group generated by $M_{1}$ and $M_{2}$ belongs to $\mathcal{M}$ since its centralizer in $G$ is equal to $C_{G}(M_{1})\cap C_{G}(M_{2})$, and the group $G/(C_{G}(M_{1})\cap C_{G}(M_{2}))$ diagonally injects into $G/C_{G}(M_{1})\times G/C_{G}(M_{2})$, which is amenable. Therefore $R_{1}$ is the union of members of $\mathcal{M}$. If $r\in R_{1}$, then $r$ is contained in some $M\in\mathcal{M}$, and since $C_{G}(M)<\bar{C}(r)$, we have $r\in R$. Assertion (iv) follows. ∎

Let $G$ be a countable group. Suppose that the AC-center of $G$, denoted by $R$, is infinite. We first assume that there exists a finite subset $S\subset R$ such that the normal closure $M\coloneqq\langle S\rangle_{G}$ is infinite. Setting $L\coloneqq C_{G}(M)$, we then have two commuting, normal subgroups $L$, $M$ of $G$ such that $M$ is amenable and the quotient group $G/(LM)$ is amenable. If $L\cap M$ is finite, then the infinite group $M/(L\cap M)$ injects into the group $(LM)/L$ and hence the index of $L$ in $LM$ is infinite. By [TD, Theorem 18 (H1)], we conclude that $G$ is stable and thus has the Schmidt property. If $L\cap M$ is infinite, then $LM$ has the infinite central subgroup $L\cap M$. Since $G/(LM)$ is amenable, the construction in the proof of [TD, Theorem 15] yields an ergodic free p.m.p. action of $G$ which is Schmidt.

We next assume that for each finite subset $S\subset R$, the normal closure $\langle S\rangle_{G}$ is finite. For each $r\in R$, the normal closure $\langle r\rangle_{G}$ is then finite. The group $G$ acts on $\langle r\rangle_{G}$ by conjugation, and some finite index subgroup of $G$ acts on it trivially. Hence the centralizer $C_{G}(r)$ is of finite index in $G$, that is, $r$ belongs to the FC-center of $G$. The AC-center $R$ is thus contained in the FC-center of $G$, and they coincide after all. Let us record the following structural alternative obtained at this point.

Proposition 3.2**.**

Let $G$ be a countable group with infinite AC-center. Then either

1. (1)

there exist two commuting, normal subgroups $L$, $M$ of $G$ such that one of them is infinite and amenable and the quotient group $G/(LM)$ is amenable, or 2. (2)

the AC-center and the FC-center of $G$ coincide, and for each finite subset of the FC-center of $G$, its normal closure in $G$ is finite.

As shown above, if there exists a finite subset $S\subset R$ such that the normal closure $\langle S\rangle_{G}$ is infinite, then case (1) occurs, and if there exists no such $S$, then case (2) occurs. In case (1), it has already shown that $G$ has the Schmidt property. Therefore for the proof of Theorem 1.1, it remains to show that $G$ has the Schmidt property if $G$ has infinite FC-center and every finite subset of the FC-center has finite normal closure in $G$.

Finally we point out the following permanence properties, which are concerned with the question in Remark 1.4, but are not necessary for the proof of Theorem 1.1.

Proposition 3.3**.**

Let $G$ be a countable group with a finite central subgroup $Z$. Then

1. (i)

the group $G$ has infinite FC-center if and only if $G/Z$ has infinite FC-center. 2. (ii)

The group $G$ has infinite AC-center if and only if $G/Z$ has infinite AC-center.

Proof.

For each $g\in G$, let $A_{G}(g)$ denote the conjugacy class of $g$ in $G$. We note that an element $g\in G$ belongs to the FC-center of $G$ if and only if the set $A_{G}(g)$ is finite. We set $\Gamma=G/Z$ with $\pi\colon G\to\Gamma$ the quotient map. Let $R^{0}$ be the FC-center of $G$ and $R_{1}^{0}$ the FC-center of $\Gamma$. For each $g\in G$, the map $\pi$ is a surjection from $A_{G}(g)$ onto $A_{\Gamma}(\pi(g))$, and is finite-to-one since $Z$ is finite. This implies that $\pi(R^{0})=R_{1}^{0}$, and assertion (i) follows.

We prove assertion (ii). Let $R$ be the AC-center of $G$ and $R_{1}$ the AC-center of $\Gamma$. It suffices to show that $\pi(R)=R_{1}$. For each $g\in G$, we have $\pi(C_{G}(\langle g\rangle_{G}))<C_{\Gamma}(\langle\pi(g)\rangle_{\Gamma})$. We thus have the surjection from $G/C_{G}(\langle g\rangle_{G})$ onto $\Gamma/C_{\Gamma}(\langle\pi(g)\rangle_{\Gamma})$. Hence $\pi(R)<R_{1}$.

We fix $\gamma\in\Gamma$ and set $M=\langle\gamma\rangle_{\Gamma}$ and $L=C_{\Gamma}(M)$. We choose a section $\mathsf{s}\colon\Gamma\to G$ of $\pi$. Let $\mathrm{Hom}(M,Z)$ be the group of homomorphisms from $M$ into $Z$ such that the product of two elements $\tau_{1},\tau_{2}\in\mathrm{Hom}(M,Z)$ is given by the homomorphism $m\mapsto\tau_{1}(m)\tau_{2}(m)$. Since $L$ and $M$ commute, we obtain the homomorphism $\tau\colon L\to\mathrm{Hom}(M,Z)$ defined by $\tau_{l}(m)=[\mathsf{s}(l),\mathsf{s}(m)]$ for $l\in L$ and $m\in M$. We set $L_{1}=\ker\tau$. Then $L/L_{1}$ is abelian and hence amenable. If $g\in G$ with $\pi(g)=\gamma$, then $L_{1}<\pi(C_{G}(\langle g\rangle_{G}))$ because for each $l\in L_{1}$, we have $\mathsf{s}(l)\in C_{G}(\mathsf{s}(M))=C_{G}(\langle g\rangle_{G})$ and $l=\pi(\mathsf{s}(l))\in\pi(C_{G}(\langle g\rangle_{G}))$.

Suppose that $\gamma\in R_{1}$ and pick $g\in G$ with $\pi(g)=\gamma$. We show that $g\in R$, which implies the inclusion $R_{1}<\pi(R)$. We set $N=C_{G}(\langle g\rangle_{G})$. The group $G/N$ is isomorphic to $\Gamma/\pi(N)$ via $\pi$. Since $L_{1}<\pi(N)$, we have the surjection from $\Gamma/L_{1}$ onto $\Gamma/\pi(N)$, which surjects onto $\Gamma/L$ because $\pi(N)<L$. It follows from $\gamma\in R_{1}$ that $\Gamma/L$ is amenable. Since $L/L_{1}$ is also amenable, so are $\Gamma/L_{1}$, $\Gamma/\pi(N)$ and $G/N$, and thus $g\in R$. ∎

3.2. An outline of Sections 4 and 5

Let $G$ be a countable group with infinite FC-center $R$. Suppose that every finite subset of $R$ has finite normal closure in $G$. The proof of the Schmidt property of $G$ will be given throughout Sections 4 and 5. In this subsection, we outline the proof along with a preliminary lemma on structure of $R$.

In Section 4, we show that $G$ has the Schmidt property under the assumption that the center of $R$ is finite. If we set $N=\bigcap_{r\in R}C_{G}(r)$, then $N\cap R$ is the center of $R$. Since $C_{G}(r)$ is of finite index in $G$ for all $r\in R$, the group $G/N$ is residually finite and thus admits a free profinite action. Moreover $G/N$ has infinite FC-center because the FC-center of $G/N$ contains $(RN)/N$. Following Popa-Vaes [PV, Theorem 6.4] and Deprez-Vaes [DV, Section 3], we construct a free profinite Schmidt action $G/N\curvearrowright(X,\mu)$ (after passing to some finite index subgroup of $G$). We then apply Theorems 2.5 and 2.14 to the translation groupoid $G\ltimes(X,\mu)$ and conclude that $G$ has the Schmidt property. We remark that the proof in Section 4 does not use the condition that every finite subset of $R$ has finite normal closure in $G$.

In Section 5, we assume that the center of $R$ is infinite. We then have an infinite abelian subgroup $A<R$ normalized by $G$. This subgroup $A$ will appropriately be chosen and is not necessarily the center of $R$. Since each finite subset of $R$ has finite normal closure in $G$, there exists a strictly increasing sequence $A_{1}<A_{2}<\cdots$ of finite subgroups of $A$ such that each $A_{n}$ is normalized by $G$. Let us draw our attention to the following condition:

1. $(\star)$

For every $N\in\mathbb{N}$, we have $\lim_{n}|F_{n,N}|/|A_{n}|=1$, where $F_{n,N}$ is the set of elements of $A_{n}$ whose order is more than $N$.

For example, if $A_{n}=\mathbb{Z}/2^{n}\mathbb{Z}$ and we embed $A_{n}$ into $A_{n+1}$ arbitrarily, then the sequence $A_{1}<A_{2}<\cdots$ fulfills this condition. In Subsection 5.3, we assume condition $(\star)$ and show that $G$ has the Schmidt property. In Subsection 5.4, we deal with the case where condition $(\star)$ is not fulfilled. In this case, applying Lemma 3.4 below, after replacing $(A_{n})$, we may assume without loss of generality that for some prime number $p$, each $A_{n}$ is isomorphic to the direct sum of copies of $\mathbb{Z}/p\mathbb{Z}$.

Lemma 3.4**.**

Let $G$ be a countable group and $A$ an infinite abelian normal subgroup of $G$ contained in the FC-center of $G$. Suppose that each finite subset of $A$ has finite normal closure in $G$ and let $A_{1}<A_{2}<\cdots$ be a strictly increasing sequence of finite subgroups of $A$ such that each $A_{n}$ is normalized by $G$. Suppose further that for this sequence, condition $(\star)$ does not hold. Then there exist a prime number $p$ and a strictly increasing sequence $B_{1}<B_{2}<\cdots$ of finite subgroups of $A$ such that each $B_{n}$ is normalized by $G$ and isomorphic to the direct sum of copies of $\mathbb{Z}/p\mathbb{Z}$.

Proof.

Since condition $(\star)$ does not hold, after passing to a subsequence of $(A_{n})$, we may assume that there exists $N\in\mathbb{N}$ such that the ratio $|A_{n}\setminus F_{n}|/|A_{n}|$ is uniformly positive, where $F_{n}$ denotes the set of elements of $A_{n}$ whose order is more than $N$. Let $\mathcal{P}$ be the set of prime numbers. Then $A_{n}$ is isomorphic to the direct sum $\bigoplus_{p\in\mathcal{P}}A_{n}^{p}$, where $A_{n}^{p}$ is the subgroup of elements of $A_{n}$ whose order is a power of $p$. This direct sum decomposition is canonical and is thus preserved under $G$-conjugation. We aim to show that for some $p\in\mathcal{P}$, the number of elements of $A_{n}^{p}$ whose order is $p$ diverges to infinity after passing to a subsequence of $(A_{n})$.

Let $C_{n}^{p}$ be the set of elements of $A_{n}^{p}$ whose order is less than or equal to $N$. Then $C_{n}^{p}$ is a subgroup of $A_{n}^{p}$. We claim that for some $p\in\mathcal{P}$, after passing to a subsequence of $(A_{n})$, we have $|C_{n}^{p}|\to\infty$ as $n\to\infty$. Otherwise for each $p\in\mathcal{P}$, the sequence $(|C_{n}^{p}|)_{n\in\mathbb{N}}$ would be bounded. Therefore $|C_{n}^{p}|$ is uniformly bounded among all $n$ and all $p\in\mathcal{P}$ with $p\leq N$. This is absurd with the condition that $|A_{n}\setminus F_{n}|/|A_{n}|$ is uniformly positive and $|A_{n}|\to\infty$, because each element of $A_{n}$ whose order is less than or equal to $N$ is a sum of elements of $C_{n}^{p}$ with $p\leq N$.

Since $C_{n}^{p}$ is isomorphic to a direct sum of groups $\mathbb{Z}/p^{k}\mathbb{Z}$ for some positive integers $k$ with $p^{k}\leq N$, it follows from $|C_{n}^{p}|\to\infty$ that the number of elements of $C_{n}^{p}$ whose order is $p$ diverges to infinity. This is the claim that we aim to show. Note that elements of $A$ of order $p$ are preserved under $G$-conjugation. Note also that each finite set of elements of $A$ of order $p$ generates a group whose elements other than the trivial one have order $p$, which is isomorphic to the direct sum of finitely many copies of $\mathbb{Z}/p\mathbb{Z}$. Hence we obtain a desired sequence $B_{1}<B_{2}<\cdots$ of subgroups inductively as follows: Choose an element of $\bigcup_{n}A_{n}$ of order $p$ and let $B_{1}$ be its normal closure in $G$. Having defined $B_{n}$, choose an element $a$ of $\bigcup_{n}A_{n}$ of order $p$ which does not belong to $B_{n}$ and let $B_{n+1}$ be the normal closure of $B_{n}\cup\{a\}$ in $G$. ∎

3.3. Examples

We present examples of groups with infinite FC-center such that their Schmidt property does not follow from known results in [PV] and [KTD] immediately. Let us recall those results:

1. (1)

If a countable group $G$ has infinite FC-center and is residually finite, then $G$ has the Schmidt property ([PV, Theorem 6.4], see also [KTD, Example 8.10]). 2. (2)

Suppose that a countable group $\Gamma$ acts on a countably infinite amenable group $A$ by automorphisms and suppose further that each $\Gamma$-orbit in $A$ is finite. Then the semi-direct product $\Gamma\ltimes A$ is stable ([KTD, Example 8.11]) and therefore has the Schmidt property.

Here we recall that a free ergodic p.m.p. action of a countable group is called stable if the associated orbit equivalence relation absorbs the ergodic p.m.p. hyperfinite equivalence relation on an atomless standard probability space, under direct product. If a countable group $G$ admits a free ergodic p.m.p. action which is stable, then $G$ is called stable.

Example 3.5*.*

Let $\Gamma$ be the group of Ershov [Er]. This is a countable, residually finite group with property (T) whose FC-center $R$ is not virtually abelian (note that these conditions imply $R\neq\Gamma$. Otherwise $R=\Gamma$ would be amenable by Lemma 3.1 (iii) and hence finite by property (T) of $\Gamma$, but this is absurd with $R$ being not virtually abelian). Let $H$ be a countable, non-residually-finite group and define $G$ as the amalgamated free product $G=\Gamma\ast_{R}(H\times R)$, where $R$ is identified with the subgroup $\{e\}\times R$ of $H\times R$. Then the FC-center of $G$ is equal to $R$, which is proved in the next paragraph, and $G$ is not residually finite. Moreover $G$ is not stable as shown in Corollary 3.10 below.

We prove that the FC-center of $G$ is equal to $R$. Pick $r\in R$. We naturally identify $H$ with the subgroup $H\times\{e\}$ of $H\times R$. Let $p\colon G\to\Gamma$ be the surjection onto the first factor. Then $\ker p=\langle H\rangle_{G}$. Since $R$ is a normal subgroup of $G$, it follows from $H<C_{G}(R)$ that $\ker p<C_{G}(R)<C_{G}(r)$. On the other hand, since $p$ is the identity on $\Gamma$, $G$ is identified with the semi-direct product $\Gamma\ltimes\ker p$. Then $C_{G}(r)$ is identified with $C_{\Gamma}(r)\ltimes\ker p$, which is of finite index in $\Gamma\ltimes\ker p$. Thus $r$ belongs to the FC-center of $G$. We have shown that $R$ is contained in the FC-center of $G$. The converse inclusion holds because the quotient group $G/R$ is isomorphic to the free product $(\Gamma/R)\ast H$ whose FC-center is trivial.

Example 3.6*.*

We set $\Gamma=\mathit{SL}_{m}(\mathbb{Z})$ with $m\geq 2$. The group $\mathbb{Z}[1/2]/\mathbb{Z}$ is identified with the increasing union $\bigcup_{n}\mathbb{Z}/2^{n}\mathbb{Z}$, where the element $1\in\mathbb{Z}/2^{n}\mathbb{Z}$ is identified with the element $1/2^{n}+\mathbb{Z}\in\mathbb{Z}[1/2]/\mathbb{Z}$. We set $A_{n}=(\mathbb{Z}/2^{n}\mathbb{Z})^{m}$ and $A=(\mathbb{Z}[1/2]/\mathbb{Z})^{m}=\bigcup_{n}A_{n}$. The group $\Gamma$ acts on each $A_{n}$ by automorphisms, and the increasing sequence $A_{1}<A_{2}<\cdots$ fulfills condition $(\star)$ in Subsection 3.2.

The semi-direct product $\Gamma\ltimes A$ is not residually finite. In fact, the group $\mathbb{Z}[1/2]/\mathbb{Z}$ has no finite index subgroup other than itself, which is proved as follows: Let $B$ be a finite index subgroup of $\mathbb{Z}[1/2]/\mathbb{Z}$ and pick $r\in\mathbb{Z}[1/2]$. Find $m\in\mathbb{N}$ with $2^{m}r\in\mathbb{Z}$. Since $B$ is of finite index, there exist $k,l\in\mathbb{N}$ such that $2^{-k}r-2^{-l}r+\mathbb{Z}\in B$ and $k-l>m$. Then the element $2^{m+l}(2^{-k}r-2^{-l}r)+\mathbb{Z}=2^{m+l-k}r+\mathbb{Z}$ belongs to $B$ and so does $r+\mathbb{Z}$. Thus we have $B=\mathbb{Z}[1/2]/\mathbb{Z}$.

Let $E$ be a countable group with property (T) containing $A$ as a central subgroup. We define $G$ as the amalgamated free product $G=(\Gamma\ltimes A)\ast_{A}E$. Then the FC-center of $G$ is equal to $A$, and $G$ is not stable (Corollary 3.10).

We obtain such a group $E$ as follows, relying on the construction of Cornulier [C] (see Appendix A for construction of analogous groups): Let $H$ be the subgroup of $\mathit{SL}_{5}(\mathbb{Z}[1/2])$ consisting of matrices of the form

[TABLE]

where $h$ runs through elements of $\mathit{SL}_{3}(\mathbb{Z}[1/2])$. Then $H$ has property (T) ([C, Proposition 2.7]). The center $C$ of $H$ consists of matrices such that each diagonal entry is $1$ and the $(1,5)$-entry is the only off-diagonal entry that is possibly non-zero. Let $Z$ be the subgroup of $C$ consisting of matrices whose $(1,5)$-entry belongs to $\mathbb{Z}$. Then the group $E\coloneqq(H/Z)^{m}$ is a desired one. Indeed $(C/Z)^{m}$ is a central subgroup of $E$ isomorphic to $A$, and $E$ has property (T) since $H$ has property (T).

Example 3.7*.*

Let $p$ be a prime number and set $A=\bigoplus_{\mathbb{N}}\mathbb{Z}/p\mathbb{Z}$. For $n\in\mathbb{N}$, we define $A_{n}$ as the group of elements $(a_{i})_{i\in\mathbb{N}}\in A$ such that $a_{i}=0$ if $i>n$. Every non-trivial element of $A$ has order $p$. Thus the increasing sequence $A_{1}<A_{2}<\cdots$ does not fulfill condition $(\star)$ in Subsection 3.2. Let $\mathcal{N}$ be the group of matrices $(a_{ij})_{i,j\in\mathbb{N}}$ with coefficient in $\mathbb{Z}/p\mathbb{Z}$ such that $a_{ii}=1$ for all $i\in\mathbb{N}$ and $a_{ij}=0$ for all $i>j$. The group $\mathcal{N}$ acts on the vector space $A$ by linear automorphisms, preserving the subspace $A_{n}$. We equip $\mathcal{N}$ with the topology of pointwise convergence as automorphisms of $A$. Then $\mathcal{N}$ is a compact group.

Let $\Gamma$ be a countable dense subgroup of $\mathcal{N}$. In the paragraph after next, we will prove that the FC-center of the semi-direct product $\Gamma\ltimes A$ is equal to $A$. As in Example 3.6, let $E$ be a countable group with property (T) containing $A$ as a central subgroup, and define $G$ as the amalgamated free product $G=(\Gamma\ltimes A)\ast_{A}E$. Then the FC-center of $G$ is equal to $A$, and $G$ is not stable (Corollary 3.10).

We find such a group $E$, relying on the construction of Cornulier [C] again: Let $\mathbb{F}_{p}$ be the field of order $p$ and let $\mathbb{F}_{p}[t]$ be the ring of polynomials over $\mathbb{F}_{p}$ in one indeterminate $t$. We define $E$ as the subgroup of $\mathit{SL}_{5}(\mathbb{F}_{p}[t])$ consisting of matrices of the form (3.1) with $h$ running through elements of $\mathit{SL}_{3}(\mathbb{F}_{p}[t])$. Then $E$ has property (T) by [C, Lemma 2.2]. The center of $E$ is isomorphic to $\mathbb{F}_{p}[t]$ and to $A$.

Let $R$ be the FC-center of $\Gamma\ltimes A$. We prove that $R$ is equal to $A$. For each $n$, the group of elements of $\Gamma$ acting on $A_{n}$ trivially is of finite index in $\Gamma$. Thus $A_{n}<R$ and $A<R$. For the converse inclusion, it suffices to show that if an element $g\in\Gamma$ centralizes a finite index subgroup of $\Gamma$, then $g$ is trivial. Suppose otherwise toward a contradiction. Write $g=(g_{ij})_{i,j\in\mathbb{N}}$ as a matrix and pick positive integers $k<l$ such that $g_{kl}\neq 0$ and $g_{kj}=0$ if $1<j<l$. Since $\Gamma$ is dense in $\mathcal{N}$ and $g$ commutes with some finite index subgroup of $\Gamma$, there exists an open neighborhood $V$ of the identity in $\mathcal{N}$ such that $g$ commutes with each element of $V$. Then there exists an $m\in\mathbb{N}$ such that if a matrix $h=(h_{ij})_{i,j}\in\mathcal{N}$ satisfies $h_{ij}=0$ for all $1\leq i<j<m$, then $h$ belongs to $V$. We may assume that $m>l$. Let $h\in V$ be the matrix such that the $(l,m)$-entry is $1$ and the other off-diagonal entries are [math]. Then the $(k,m)$-entries of $gh$ and $hg$ are $g_{kl}+g_{km}$ and $g_{km}$, respectively. We thus have $gh\neq hg$, a contradiction.

We present a sufficient condition for a countable group not to be stable, and apply it to the groups in the above examples. We say that a mean on a countable group $G$ is diffuse if its value on each finite subset of $G$ is zero.

Proposition 3.8**.**

Let $G$ be a countable group and $A$ a subgroup of $G$. Suppose that each diffuse, $G$-conjugation invariant mean on $G$ is supported on $A$ and that the pair $(G,A)$ has property (T). Then $G$ is not stable.

Proof.

Suppose that $G$ admits a free ergodic p.m.p. action $G\curvearrowright(X,\mu)$ which is stable. Then we have a central sequence $(T_{n})$ in the full group $[G\ltimes(X,\mu)]$ and an asymptotically invariant sequence $(A_{n})$ for $G\ltimes(X,\mu)$ such that $T_{n}^{\circ}A_{n}\cap A_{n}=\emptyset$ (and hence $\mu(A_{n})=1/2$) for all $n$ (see Remark 3.9 below). Property (T) of the pair $(G,A)$ implies that there exists an $A$-invariant Borel subset $B_{n}\subset X$ such that $\mu(A_{n}\bigtriangleup B_{n})\to 0$. Since the functions on $G$ defined by $g\mapsto\mu(\{\,x\in X\mid T_{n}x=g\,\})$ are asymptotically $G$-conjugation invariant, the assumption on $G$-conjugation invariant means on $G$ implies that there exists a Borel subset $D_{n}\subset X$ such that $T_{n}x\in A$ for all $x\in D_{n}$ and $\mu(D_{n})\to 1$. Then

[TABLE]

where the last equation holds since $B_{n}$ is $A$-invariant and $T_{n}x\in A$ for all $x\in D_{n}$. Thus $\mu(T_{n}^{\circ}B_{n}\bigtriangleup B_{n})\leq 2\mu(X\setminus D_{n})\to 0$ and $\mu(T_{n}^{\circ}A_{n}\bigtriangleup A_{n})\to 0$, a contradiction. ∎

Remark 3.9*.*

Let the group $\bigoplus_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$ act on the compact group $X_{0}=\prod_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$ by translation, equip $X_{0}$ with the Haar measure, and let $\mathcal{R}_{0}$ denote the associated orbit equivalence relation. For each $n\in\mathbb{N}$, let $\bar{T}_{n}\in[\mathcal{R}_{0}]$ be the element of $\bigoplus_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$ such that its coordinate indexed by $n$ is $1$ and the other coordinates are [math], and let $\bar{A}_{n}\subset X_{0}$ be the subset consisting of points whose coordinate indexed by $n$ is [math]. Then $(\bar{T}_{n})$ is central in $[\mathcal{R}_{0}]$, $(\bar{A}_{n})$ is asymptotically invariant for $\mathcal{R}_{0}$, and $\bar{T}_{n}\bar{A}_{n}\cap\bar{A}_{n}=\emptyset$ for all $n$.

If a discrete p.m.p. equivalence relation $\mathcal{R}$ is stable, then we obtain similar sequences as follows: By stability, we have a decomposition $\mathcal{R}=\mathcal{R}_{0}\times\mathcal{R}_{1}$, where $\mathcal{R}_{1}$ is some discrete p.m.p. equivalence relation on a standard probability space $(X_{1},\mu_{1})$. Define $T_{n}\in[\mathcal{R}]$ by $T_{n}(x,y)=(\bar{T}_{n}(x),y)$ for $x\in X_{0}$ and $y\in X_{1}$, and set $A_{n}=\bar{A}_{n}\times X_{1}$. Then $(T_{n})$ is central in $[\mathcal{R}]$, $(A_{n})$ is asymptotically invariant for $\mathcal{R}$, and $T_{n}A_{n}\cap A_{n}=\emptyset$ for all $n$.

Corollary 3.10**.**

None of the groups $G$ in Examples 3.5–3.7 is stable.

Proof.

Let $G=\Gamma\ast_{R}(H\times R)$ be the group in Example 3.5. Then $G$ surjects onto the free product $(\Gamma/R)\ast H$ with kernel $R$. Since each conjugation-invariant mean on $(\Gamma/R)\ast H$ is supported on the trivial element ([BH, Théorème 5 (c)]), each $G$-conjugation invariant mean on $G$ is supported on $R$. Since $\Gamma$ has property (T), so does the pair $(G,R)$. Thus Proposition 3.8 applies.

Let $G=(\Gamma\ltimes A)\ast_{A}E$ be the group in Example 3.6 or 3.7. It similarly turns out that each $G$-conjugation invariant mean on $G$ is supported on $A$. Since $E$ has property (T), so does the pair $(G,A)$. Thus Proposition 3.8 applies. ∎

Remark 3.11*.*

Let $\Gamma$ be a countable group acting on a countably infinite amenable group $A$ by automorphisms. The semi-direct product $G\coloneqq\Gamma\ltimes A$ then acts on $A$ by affine transformations, i.e., $\Gamma$ acts on $A$ by automorphisms, and $A$ acts on $A$ by left multiplication. If the action of $G$ on $A$ admits an invariant mean, then the pair $(G,A)$ does not have property (T). Indeed, the associated unitary representation of $G$ on $\ell^{2}(A)$ weakly contains the trivial representation, but has no $A$-invariant unit vector.

If each $\Gamma$-orbit in $A$ is finite, then the action of $G$ on $A$ admits an invariant mean (see the proof of [TD, Theorem 13, ii]). Therefore for the stable group $G=\Gamma\ltimes A$ reviewed in the beginning of this subsection, the pair $(G,A)$ does not have property (T). We refer to [DV, Proposition 3.1], [Ki3, Theorem 1.1] and [TD, 0.H] for other relationships between stability and relative property (T).

4. Groups with non-commutative FC-center

Let $G$ be a countable group with infinite FC-center $R$. Suppose that the center of $R$ is finite. In this section, we aim to prove that $G$ has the Schmidt property.

We set $N=\bigcap_{r\in R}C_{G}(r)$. Then $R$ and $N$ commute and $N\cap R$ is exactly the center of $R$. We may assume without loss of generality that $N\cap R$ is central in $G$ after passing to some finite index subgroup of $G$. Indeed the subgroup $G_{0}\coloneqq\bigcap_{r\in N\cap R}C_{G}(r)$ is of finite index in $G$ since $N\cap R$ is finite, and $G_{0}$ commutes with $N\cap R$. Since $N\cap R$ is central in $R$, we have $R<G_{0}$ and hence the FC-center of $G_{0}$ is equal to $R$. If we set $N_{0}=\bigcap_{r\in R}C_{G_{0}}(r)$, then $N_{0}=N\cap G_{0}$ and hence $N_{0}\cap R$ is finite and central in $G_{0}$. In general for a finite index inclusion $\Lambda<\Gamma$ of countable groups, if $\Lambda$ admits a free ergodic p.m.p. action which is Schmidt, then the action of $\Gamma$ induced (not co-induced) from it is also Schmidt. Therefore after replacing $G$ with $G_{0}$, we may assume that $N\cap R$ is central in $G$.

Let $G=H_{0}>H_{1}>H_{2}>\cdots$ be a decreasing sequence of finite index subgroups of $G$ such that $\bigcap_{n}H_{n}=N$. We can choose a sequence $(r_{n})_{n\in\mathbb{N}}$ of elements of $R\setminus N$ such that

1. (i)

if $n\neq m$, then $r_{n}$ and $r_{m}$ are distinct in the quotient group $R/(N\cap R)$, and 2. (ii)

for each $n\in\mathbb{N}$, $r_{n}$ belongs to $C_{G}(r_{1},\ldots,r_{n-1})\cap H_{n-1}$.

Indeed we first note that $R/(N\cap R)$ is infinite since $R$ is infinite and $N\cap R$ is finite. Let $r_{1}$ be an arbitrary element of $R\setminus N$. If $r_{1},\ldots,r_{n-1}$ are chosen, then $C_{G}(r_{1},\ldots,r_{n-1})\cap H_{n-1}$ is of finite index in $G$ and hence its image in $G/(N\cap R)$ is of finite index. The intersection of that image with $R/(N\cap R)$ is of finite index in $R/(N\cap R)$ and hence infinite. If we let $r_{n}$ be an element of $R\setminus N$ whose image in $R/(N\cap R)$ belongs to that intersection and is distinct from the images of $r_{1},\ldots,r_{n-1}$, then conditions (i) and (ii) are fulfilled. For an integer $n\geq 2$, we set

[TABLE]

Let $G\curvearrowright(X,\mu)$ be the ergodic p.m.p. action obtained as the inverse limit of the system of the p.m.p. actions $G\curvearrowright G/G_{n}$ given by left multiplication. Then $N$ acts on $X$ trivially, and the induced action $G/N\curvearrowright(X,\mu)$ is free because $\bigcap_{n}H_{n}=N$.

We show that the translation groupoid $(\mathcal{G},\mu)\coloneqq G\ltimes(X,\mu)$ admits a central sequence $(T_{n})$ in its full group such that $T_{n}^{\circ}x\neq x$ and $T_{n}x\in R$ for all $n$ and all $x\in X$. Let $p_{n}\colon X\to G/G_{n}$ be the projection obtained from the inverse limit construction. We define a map $T_{n}\colon X\to G$ by $T_{n}x=gr_{n}g^{-1}$ for $x\in p_{n}^{-1}(gG_{n})$ and $g\in G$. This does not depend on the choice of $g$ because $r_{n}$ commutes with every element of $G_{n}$ by the definition of $G_{n}$. Since $r_{n}$ belongs to $G_{n}$ by condition (ii), $T_{n}^{\circ}$ preserves the subset $p_{n}^{-1}(gG_{n})$ for each $g\in G$. Therefore $T_{n}$ belongs to $[\mathcal{G}]$ and we have $\mu(T_{n}^{\circ}A\bigtriangleup A)\to 0$ for every Borel subset $A\subset X$. For each $h\in G$, $T_{n}$ commutes with the element $\phi_{h}\in[\mathcal{G}]$ defined as the constant map with value $h$. Indeed if $x\in p_{n}^{-1}(gG_{n})$ with $g\in G$, then $(T_{n}\circ\phi_{h})x=T_{n}(hx)h=hgr_{n}g^{-1}$, which is equal to $(\phi_{h}\circ T_{n})x$. Therefore $(T_{n})$ is a central sequence in $[\mathcal{G}]$, and we have $T_{n}^{\circ}x\neq x$ for every $x\in X$ because $r_{n}$ does not belong to $N$.

We thus obtained the ergodic p.m.p. action $G\curvearrowright(X,\mu)$ such that $N$ acts on $X$ trivially, the induced action of $G/N$ on $X$ is free, and there exists a central sequence $(T_{n})$ in the full group $[G\ltimes(X,\mu)]$ such that $T_{n}x\neq x$ and $T_{n}x\in R$ for all $n$ and all $x\in X$. Recall also that $R$ is contained in the centralizer $C_{G}(N)$ and that $N\cap R$ is finite and central in $G$. In order to apply Theorem 2.5 or 2.14, we check that at least one of the assumptions in those two theorems is fulfilled. For $p\in\mathbb{N}$, let $A_{n}^{p}\subset X$ be the set of $p$-periodic points of $T_{n}^{\circ}$. If every $p\in\mathbb{N}$ satisfies $\mu(A_{n}^{p})\to 0$ as $n\to\infty$, then letting $\Omega$ be a singleton and $M_{\omega}=N$ in Theorem 2.5, we apply it and conclude the Schmidt property for $G$. Suppose otherwise, i.e., suppose that for some integer $p\geq 2$, the measure $\mu(A_{n}^{p})$ does not converge to [math] as $n\to\infty$. After passing to a subsequence, we may assume that $\mu(A_{n}^{p})$ is uniformly positive. If $x\in A_{n}^{p}$, then $(T_{n}^{\circ})^{p}x=x$ and hence $(T_{n})^{p}x\in N$ and $(T_{n})^{p}x\in N\cap R$. Letting $M=N$ and $L=N\cap R$ in Theorem 2.14, we apply it and conclude the Schmidt property of $G$.

5. Groups with commutative FC-center

5.1. Groupoid extensions

Let $G$ be a countable group and let $A$ be an abelian normal subgroup of $G$. We set $\Gamma=G/A$ and choose a section $\mathsf{s}\colon\Gamma\to G$ of the quotient map, with $\mathsf{s}(e)=e$. We then have the 2-cocycle $\sigma\colon\Gamma\times\Gamma\to A$ defined by $\sigma(g,h)\mathsf{s}(gh)=\mathsf{s}(g)\mathsf{s}(h)$ for $g,h\in\Gamma$. The map $\sigma$ satisfies the 2-cocycle identity

[TABLE]

for all $g,h,k\in\Gamma$, where we set ${}^{g}a=\mathsf{s}(g)a\mathsf{s}(g)^{-1}$ for $g\in\Gamma$ and $a\in A$. Note that ${}^{g}a$ does not depend on the choice of the section $\mathsf{s}$.

Fix a compact abelian metrizable group $L$. We define $X$ as the group of homomorphisms from $A$ into $L$, identified with the closed subgroup of the product group $\prod_{A}L$. Let $\mu$ denote the normalized Haar measure on $X$. The group $G$ acts on $X$ by $(g\tau)(a)=\tau(g^{-1}ag)$ for $g\in G$, $a\in A$ and $\tau\in X$, and this gives rise to the action of $\Gamma$ on $X$. We set $\mathcal{U}=X\times L$ and regard it as the bundle over $X$ with respect to the projection onto the first coordinate. We also regard $\mathcal{U}$ as the groupoid with unit space $X$ such that the range and source maps are the projection onto $X$, and the product is given by $(\tau,l)(\tau,m)=(\tau,lm)$ for $\tau\in X$ and $l,m\in L$. The translation groupoid $X\rtimes\Gamma$ acts on $\mathcal{U}$ by $(\tau,g)(g^{-1}\tau,l)=(\tau,l)$ for $\tau\in X$, $g\in\Gamma$ and $l\in L$.

Let $(X\rtimes\Gamma)^{(2)}$ be the set of composable pairs of the groupoid $X\rtimes\Gamma$, i.e., the set of all pairs of the form $((\tau,g),(g^{-1}\tau,h))$ for some $\tau\in X$ and $g,h\in\Gamma$. The pair of that form is also denoted by $(\tau,g,h)$ for brevity. We define the 2-cocycle $\tilde{\sigma}\colon(X\rtimes\Gamma)^{(2)}\to\mathcal{U}$ by

[TABLE]

where $\langle\tau,a\rangle$ stands for $\tau(a)$ for $\tau\in X$ and $a\in A$. Indeed the map $\tilde{\sigma}$ satisfies the 2-cocycle identity:

[TABLE]

where we set ${}^{(\tau,g)}(g^{-1}\tau,l)=(\tau,l)$ for $(\tau,g)\in X\rtimes\Gamma$ and $l\in L$, which is the result of the action of $(\tau,g)$ on $(g^{-1}\tau,l)\in\mathcal{U}$. Let us check equation (5.2): For the first coordinate in $X$, both sides are $\tau$. For the second coordinate in $L$, the left hand side is

[TABLE]

which is equal to the second coordinate of the right hand side.

We now construct the groupoid extension

[TABLE]

associated with the 2-cocycle $\tilde{\sigma}$ (see [Se] for the extension associated with a 2-cocycle of an equivalence relation with coefficient in a bundle of abelian Polish groups). As a set, we define $\mathcal{G}_{\tilde{\sigma}}$ as the fibered product $\mathcal{U}\times_{X}(X\rtimes\Gamma)$ with respect to the range map of $X\rtimes\Gamma$. The range and source of $(u,g)\in\mathcal{G}_{\tilde{\sigma}}$ with $u\in\mathcal{U}$ and $g\in X\rtimes\Gamma$ are defined as the range and source of $g$, respectively. The product of $\mathcal{G}_{\tilde{\sigma}}$ is given by

[TABLE]

for $(u,g),(v,h)\in\mathcal{G}_{\tilde{\sigma}}$ with $(g,h)$ composable. This product is associative. Indeed for three elements $(u,g),(v,h),(w,k)\in\mathcal{G}_{\tilde{\sigma}}$ with $(g,h)$ and $(h,k)$ composable, we have

[TABLE]

The inverse of an element $(u,g)\in\mathcal{G}_{\tilde{\sigma}}$ is given by

[TABLE]

where the left hand side is a left inverse of $(u,g)$, the right hand side is a right inverse of $(u,g)$, and these two coincide because it follows from $\mathsf{s}(e)=e$ that $\sigma(g,e)=e=\sigma(e,g)$ for every $g\in\Gamma$, and $\sigma(g,g^{-1})={}^{g}\sigma(g^{-1},g)$ by the 2-cocycle identity. All these groupoid operations are Borel, and we thus obtain a Borel groupoid $\mathcal{G}_{\tilde{\sigma}}$. We have the projection from $\mathcal{G}_{\tilde{\sigma}}=\mathcal{U}\times_{X}(X\rtimes\Gamma)$ onto $X\rtimes\Gamma$, whose kernel is identified with $\mathcal{U}$ via the inclusion of $\mathcal{U}$ into $\mathcal{G}_{\tilde{\sigma}}$, $(\tau,l)\mapsto((\tau,l),(\tau,e))$ for $\tau\in X$ and $l\in L$. Consequently the groupoid extension (5.3) is obtained.

An element $((\tau,l),(\tau,\gamma))\in\mathcal{G}_{\tilde{\sigma}}=\mathcal{U}\times_{X}(X\rtimes\Gamma)$ is also denoted by $(\tau,l,\gamma)$ for brevity. We define a homomorphism $\eta\colon X\rtimes G\to\mathcal{G}_{\tilde{\sigma}}$ by

[TABLE]

for $\tau\in X$, $a\in A$ and $\gamma\in\Gamma$, where $G$ is identified with $A\times\Gamma$ via the map $(a,\gamma)\mapsto a\mathsf{s}(\gamma)$. To check that $\eta$ is indeed a homomorphism, let us recall the product of two elements of $A\times\Gamma$ inherited from $G$:

[TABLE]

for $a,b\in A$ and $\gamma,\delta\in\Gamma$. If we put $g=(a,\gamma)$ and $h=(b,\delta)$ and regard them as elements of $G$, then for each $\tau\in X$, we have

[TABLE]

where in the fourth term, the element of $\mathcal{G}_{\tilde{\sigma}}$ is written as a pair of an element of $\mathcal{U}$ and an element of $X\rtimes\Gamma$. Therefore $\eta$ is a homomorphism. The kernel of $\eta$ is given by

[TABLE]

The image of $X\rtimes A$ under $\eta$ is given by

[TABLE]

5.2. A free action from co-induction

We keep the notation in the previous subsection, where we constructed the groupoid $\mathcal{G}_{\tilde{\sigma}}$. In this subsection, we construct a free p.m.p. action of $\mathcal{G}_{\tilde{\sigma}}$, which will be obtained as the action co-induced from the shift action of $\mathcal{U}$ onto itself. This action was not treated in Subsection 2.3 since $\mathcal{G}_{\tilde{\sigma}}$ is not necessarily discrete. We do not aim to discuss co-induced actions for non-discrete Borel groupoids in full generality.

We set $\mathcal{G}=\mathcal{G}_{\tilde{\sigma}}$ and $\mathcal{Q}=X\rtimes\Gamma$ for brevity. We have the groupoid extension

[TABLE]

Recall that $\mathcal{U}=X\times L$ is the bundle of a compact abelian metrizable group $L$, and denote by $\mathcal{U}_{x}$ the fiber of $\mathcal{U}$ at $x$, i.e., $\{x\}\times L$. Each fiber $\mathcal{U}_{x}$ is often identified with $L$ naturally if there is no cause of confusion. The bundle $\mathcal{U}$ is a groupoid on $X$ and acts on itself by left multiplication. We co-induce this action to the action of $\mathcal{G}$ in the same manner as in Subsection 2.3 as follows: For $x\in X$, we set

[TABLE]

and define $Z$ as the disjoint union $Z=\bigsqcup_{x\in X}Z_{x}$. For each $f\in Z_{x}$, it is natural to regard the value $f(g)\in L$ at $g\in\mathcal{G}^{x}$ as an element of $\mathcal{U}_{s(g)}$. The set $Z$ is fibered with respect to the projection $p\colon Z\to X$ sending each element of $Z_{x}$ to $x$. Then $\mathcal{G}$ acts on $Z$ by

[TABLE]

for $g\in\mathcal{G}_{x}$, $h\in\mathcal{G}^{r(g)}$ and $f\in Z_{x}$ with $x\in X$.

We define a measure-space structure on $Z$. Recall that as a set, $\mathcal{G}$ is the fibered product $\mathcal{U}\times_{X}\mathcal{Q}$ with respect to the range map of $\mathcal{Q}$. For $\gamma\in\Gamma$, we define a map $\psi_{\gamma}\colon X\to\mathcal{G}$ by $\psi_{\gamma}(x)=((x,e),(x,\gamma))$ for $x\in X$. Then for each $x\in X$, we have $\psi_{\gamma}(x)\in\mathcal{G}^{x}$ and the family $\{\psi_{\gamma}(x)\}_{\gamma\in\Gamma}$ is a complete set of representatives of all the equivalence classes in $\mathcal{G}^{x}$, where the equivalence relation on $\mathcal{G}^{x}$ is defined as follows: two elements $g,h\in\mathcal{G}^{x}$ are equivalent if and only if $g^{-1}h\in\mathcal{U}$. Then $Z$ is identified with the product space $X\times\prod_{\Gamma}L$ under the map sending each $f\in Z_{x}$ with $x\in X$ to $(x,(f(\psi_{\gamma}(x)))_{\gamma})$. The measure-space structure on $Z$ is induced by this identification, where the space $X\times\prod_{\Gamma}L$ is equipped with the product measure of $\mu$ and the normalized Haar measure on $L$. The action of $\mathcal{G}$ on $Z$ is Borel and preserves the probability measure on $Z$ in the following sense:

Proposition 5.1**.**

With the above notation,

1. (i)

for all $\gamma\in\Gamma$, $x\in X$ and $l\in L$, we have

[TABLE]

where we identify $\mathcal{U}$ with a subset of $\mathcal{G}$ under the injection of $\mathcal{U}$ into $\mathcal{G}$. 2. (ii)

We define an action of the group $L$ on $Z$ by $lf=(x,l)f$ for $l\in L$ and $f\in Z_{x}$ with $x\in X$. Then this action is Borel, p.m.p. and free. 3. (iii)

For each $\gamma\in\Gamma$, the action of $\psi_{\gamma}$ on $Z$ is Borel and p.m.p., that is, the map from $Z$ into itself sending each $f\in Z_{x}$ with $x\in X$ to $\psi_{\gamma}(\gamma x)f\in Z_{\gamma x}$ is Borel and p.m.p. 4. (iv)

Suppose that either $L$ is infinite and $|\Gamma|\geq 3$ or $L$ is non-trivial and $\Gamma$ is infinite. Then the action of $\mathcal{G}$ on $Z$ is essentially free, i.e., for almost every $f\in Z$, letting $x\in X$ be the point with $f\in Z_{x}$, we have $gf\neq f$ for each $g\in\mathcal{G}_{x}$ except for the unit at $x$.

Proof.

To prove assertion (i), we pick $\gamma\in\Gamma$, $x\in X$ and $l\in L$ and set $g=(x,\gamma)\in\mathcal{Q}$. It follows from formula (5.5) that $\psi_{\gamma}(x)^{-1}=(\tilde{\sigma}(g^{-1},g)^{-1},g^{-1})$ and therefore

[TABLE]

where the first and second equations are derived from formula (5.4). Assertion (i) follows.

We prove assertion (ii). Pick $l\in L$ and $f\in Z_{x}$ with $x\in X$. The element $f$ is identified with the element of $X\times\prod_{\Gamma}L$ given by the pair of $x$ and the function $\gamma\mapsto f(\psi_{\gamma}(x))$. Let us describe the element of $X\times\prod_{\Gamma}L$ corresponding to $lf$, which is the pair of $x$ and the function $\gamma\mapsto(lf)(\psi_{\gamma}(x))$. For each $\gamma\in\Gamma$, we have

[TABLE]

where we apply assertion (i) in the third equation. Therefore the action of $l$ on $X\times\prod_{\Gamma}L$ is given by $(x,(l_{\gamma})_{\gamma})\mapsto(x,(ll_{\gamma})_{\gamma})$, and the action of $L$ on $Z$ is Borel, p.m.p. and free.

We prove assertion (iii). Pick $\gamma\in\Gamma$ and $f\in Z_{x}$ with $x\in X$. The element $f$ is identified with the element of $X\times\prod_{\Gamma}L$ given by the pair of $x$ and the function $\delta\mapsto f(\psi_{\delta}(x))$. The element $\psi_{\gamma}(\gamma x)f$ corresponds to the pair of $\gamma x$ and the function

[TABLE]

We set $g=(\gamma x,\gamma)$ and $h=(\gamma x,\delta)$. By formula (5.5), $\psi_{\gamma}(\gamma x)^{-1}=(\tilde{\sigma}(g^{-1},g)^{-1},g^{-1})$. For each $\delta\in\Gamma$, if we define $k\in L$ by

[TABLE]

then we have

[TABLE]

where the second equation follows from formula (5.4) and the fifth equation follows from assertion (i). Therefore

[TABLE]

and the action of $\psi_{\gamma}$ on $X\times\prod_{\Gamma}L$ is given by

[TABLE]

where the element $k=k_{\gamma,\delta,x}\in L$ is determined by equation (5.6). By the definition of $\tilde{\sigma}$ in (5.1), the function $x\mapsto k_{\gamma,\delta,x}$ is Borel. Hence the action of $\psi_{\gamma}$ is Borel and also p.m.p. by the above description of the action. Assertion (iii) follows.

We prove assertion (iv). Recall that each $g\in\mathcal{G}_{x}$ with $x\in X$ is written as $(\gamma x,l,\gamma)$ for some $\gamma\in\Gamma$ and $l\in L$. By assertion (ii), it suffices to show that for each non-trivial $\gamma\in\Gamma$, there exists a conull subset $\bar{Z}\subset Z$ such that for all $f\in\bar{Z}$ and all $l\in L$, letting $x\in X$ be the point with $f\in Z_{x}$, we have $(\gamma x,l,\gamma)f\neq f$. We fix a non-trivial $\gamma\in\Gamma$. The action of $g=(\gamma x,l,\gamma)$ on $X\times\prod_{\Gamma}L$ is described as

[TABLE]

Thus if $g$ fixes the point $(x,(l_{\delta})_{\delta})$, then $lk_{\gamma,\delta,x}^{-1}l_{\gamma^{-1}\delta}=l_{\delta}$ for all $\delta\in\Gamma$.

Suppose that $L$ is infinite and $|\Gamma|\geq 3$. Pick a non-trivial element $\gamma_{1}\in\Gamma$ with $\gamma_{1}\neq\gamma^{-1}$. We fix $x\in X$. If a point $(l_{\delta})_{\delta}$ is such that for some $l\in L$, we have $lk_{\gamma,\delta,x}^{-1}l_{\gamma^{-1}\delta}=l_{\delta}$ for all $\delta\in\Gamma$, then $lk_{\gamma,e,x}^{-1}l_{\gamma^{-1}}=l_{e}$ and $lk_{\gamma,\gamma_{1},x}^{-1}l_{\gamma^{-1}\gamma_{1}}=l_{\gamma_{1}}$. Deleting $l$, we thus obtain

[TABLE]

which says that $l_{\gamma_{1}}$ is determined if $l_{e}$, $l_{\gamma^{-1}}$ and $l_{\gamma^{-1}\gamma_{1}}$ are determined. The element $\gamma_{1}$ is distinct from all of $e$, $\gamma^{-1}$ and $\gamma^{-1}\gamma_{1}$. Hence by Fubini’s theorem, the set of points $(l_{\delta})_{\delta}$ satisfying equation (5.7) is null, where we use the assumption that $L$ is infinite and thus each singleton subset of $L$ is null. Since $x$ is an arbitrary point of $X$, by Fubini’s theorem again, the set of points $(x,(l_{\delta})_{\delta})\in X\times\prod_{\Gamma}L$ satisfying equation (5.7) is null. Thus it suffices to let $\bar{Z}$ be the complement of that null set.

Suppose next that $L$ is non-trivial and $\Gamma$ is infinite. Then there exists an infinite subset $S\subset\Gamma$ such that $S$ and $\gamma^{-1}S$ are disjoint. We fix $x\in X$. Let $(l_{\delta})_{\delta}$ be a point such that for some $l\in L$, we have $lk_{\gamma,\delta,x}^{-1}l_{\gamma^{-1}\delta}=l_{\delta}$ for all $\delta\in\Gamma$. As in the previous paragraph, for all distinct $\gamma_{0},\gamma_{1}\in S$, we then have

[TABLE]

The element $\gamma_{1}$ is distinct from all of $\gamma_{0}$, $\gamma^{-1}\gamma_{0}$ and $\gamma^{-1}\gamma_{1}$. Hence by Fubini’s theorem, for all distinct $\gamma_{0},\gamma_{1}\in S$, the set of points $(l_{\delta})_{\delta}$ satisfying equation (5.8) has measure less than $1$, where we use the assumption that $L$ is non-trivial and thus each singleton subset of $L$ has measure less than $1$. Since we have mutually disjoint, infinitely many pairs of distinct elements of $S$, the set of points $(l_{\delta})_{\delta}$ satisfying equation (5.8) for all distinct $\gamma_{0},\gamma_{1}\in S$ is null. We thus obtain $\bar{Z}$ as well as before, and assertion (iv) follows. ∎

5.3. The case where condition $(\star)$ holds

Let $G$ be a countable group and let $A$ be an infinite abelian normal subgroup of $G$ contained in the FC-center of $G$. Suppose that each finite subset of $A$ has finite normal closure in $G$ and let $A_{1}<A_{2}<\cdots$ be a strictly increasing sequence of finite subgroups of $A$ such that each $A_{n}$ is normalized by $G$. Suppose further that condition $(\star)$ introduced in Subsection 3.2 holds, i.e., for all $N\in\mathbb{N}$, we have $\lim_{n}|F_{n,N}|/|A_{n}|=1$, where $F_{n,N}$ is the set of elements of $A_{n}$ whose order is more than $N$. Under these assumptions, we aim to construct a free p.m.p. Schmidt action of $G$. We may assume that $G/A$ is infinite because otherwise $G$ is amenable. This assumption will be used in applying Proposition 5.1 (iv) later, and not used for other purposes.

We set $\Gamma=G/A$ and choose a section $\mathsf{s}\colon\Gamma\to G$ of the quotient map with $\mathsf{s}(e)=e$. We then obtain the 2-cocycle $\sigma\colon\Gamma\times\Gamma\to A$. We define $X$ as the dual group $\widehat{A}$ of $A$, i.e., the group of homomorphisms from $A$ into the torus $\mathbb{T}=\{\,z\in\mathbb{C}\mid|z|=1\,\}$. Let $\mu$ be the normalized Haar measure on $X$. We recall the construction in Subsection 5.1. Define the action of $G$ on $X$ by $(g\tau)(a)=\tau(g^{-1}ag)$ for $g\in G$, $a\in A$ and $\tau\in X$, which induces the action of $\Gamma$ on $X$. Let $\mathcal{U}\coloneqq X\times\mathbb{T}$ be the bundle over $X$, which is a groupoid with unit space $X$. Then we obtain the 2-cocycle $\tilde{\sigma}\colon(X\rtimes\Gamma)^{(2)}\to\mathcal{U}$ by formula (5.1) and obtain the groupoid extension

[TABLE]

together with the homomorphism $\eta\colon X\rtimes G\to\mathcal{G}_{\tilde{\sigma}}$ such that

[TABLE]

and $\eta(\tau,a)=(\tau,\tau(a))\in\mathcal{U}$ for all $a\in A$ and $\tau\in X$.

Let $\mathcal{G}_{\tilde{\sigma}}\curvearrowright(Z,\zeta)$ be the free p.m.p. action constructed in Subsection 5.2, i.e., the action co-induced from the shift action of $\mathcal{U}$ on itself. The space $Z$ is fibered over $X$. The fiber at $\tau\in X$ is denoted by $Z_{\tau}$. For $n\in\mathbb{N}$, let $\Gamma_{n}$ be the group of elements of $\Gamma$ acting on $A_{n}$ trivially. Let $\Gamma\curvearrowright(Y,\nu)$ be the profinite p.m.p. action associated with the system of the p.m.p. action $\Gamma\curvearrowright\Gamma/\Gamma_{n}$ given by left multiplication. Through the quotient map from $\mathcal{G}_{\tilde{\sigma}}$ onto $\Gamma$ factoring through $X\rtimes\Gamma$, we obtain the p.m.p. action $\mathcal{G}_{\tilde{\sigma}}\curvearrowright(Y,\nu)$. Then $\mathcal{G}_{\tilde{\sigma}}$ acts on $Y\times Z$ diagonally, where $Y\times Z$ is fibered over $X$ with respect to the map sending each element of $Y\times Z_{\tau}$ to $\tau$ for each $\tau\in X$.

Through the homomorphism $\eta\colon X\rtimes G\to\mathcal{G}_{\tilde{\sigma}}$, we obtain the p.m.p. action of $X\rtimes G$ on the product space $(W,\omega)\coloneqq(Y\times Z,\nu\times\zeta)$. We then obtain the p.m.p. action $G\curvearrowright(W,\omega)$ given by $g(y,z)=(g\tau,g)(y,z)$ for $g\in G$, $y\in Y$ and $z\in Z_{\tau}$ with $\tau\in X$. The action of $A$ on $W$ is given by $a(y,z)=(y,(\tau,\tau(a))z)$ for each $a\in A$. Recall that we defined the action of $\mathbb{T}$ on $Z$ by $tz=(\tau,t)z$ for $t\in\mathbb{T}$ and $z\in Z_{\tau}$ with $\tau\in X$ in Proposition 5.1 (ii). Thus, with respect to this action, the element $(y,(\tau,\tau(a))z)$ is written as $(y,\tau(a)z)$.

We now construct a central sequence $(T_{N})$ in the full group of the translation groupoid $G\ltimes(W,\omega)$. Pick $N\in\mathbb{N}$. By condition $(\star)$, for some $n=n_{N}\in\mathbb{N}$, we have $|F_{n}|/|A_{n}|\geq 1-1/N$, where $F_{n}$ is the set of elements of $A_{n}$ whose order is more than $N$. Since the dual $\widehat{A}_{n}$ of $A_{n}$ is isomorphic to $A_{n}$ ([F, Corollary 4.8]), if $E_{n}$ denotes the set of elements of $\widehat{A}_{n}$ whose order is more than $N$, then $|E_{n}|/|\widehat{A}_{n}|\geq 1-1/N$. The set $E_{n}$ is further $\Gamma$-invariant. The map $p_{n}\colon X=\widehat{A}\to\widehat{A}_{n}$ induced by the inclusion of $A_{n}$ into $A$ is surjective ([F, Corollary 4.42]). For each $\tau\in E_{n}$, since its order is more than $N$, there exists $a_{\tau}\in A_{n}$ such that

[TABLE]

We define a map $T_{N}\colon W\to A$ as follows: Let $Y_{n}$ denote the inverse image of the coset $e\Gamma_{n}$ under the projection from $Y$ onto $\Gamma/\Gamma_{n}$. For $y\in gY_{n}$ with $g\in\Gamma$ and $z\in Z_{\tau}$ with $\tau\in X$, if $\tau$ belongs to $p_{n}^{-1}(E_{n})$, then we set

[TABLE]

and otherwise we set $T_{N}(y,z)=e$. This is well-defined because $\Gamma_{n}$ acts on $A_{n}$ and $\widehat{A}_{n}$ trivially. The map from $W$ into itself, $w\mapsto(T_{N}w)w$, is an automorphism of $W$ because $A$ acts on $Y$ trivially and preserves each fiber $Z_{\tau}$ with $\tau\in X$. Thus $T_{N}$ is an element of the full group $[G\ltimes(W,\omega)]$.

Lemma 5.2**.**

With the above notation,

1. (i)

for each $N\in\mathbb{N}$ and $g\in G$, we have $\phi_{g}\circ T_{N}=T_{N}\circ\phi_{g}$, where $\phi_{g}\colon X\to G$ is the element of the full group $[G\ltimes(W,\omega)]$ given by the constant map with value $g$. 2. (ii)

For each Borel subset $B\subset W$, we have $\omega(T_{N}^{\circ}B\bigtriangleup B)\to 0$ as $N\to\infty$. 3. (iii)

We define $B_{N}\subset W$ as the set of periodic points of $T_{N}^{\circ}$ whose period is more than $N$. Then $\omega(B_{N})\geq 1-1/N$ for all $N\in\mathbb{N}$.

Proof.

To prove assertion (i), we pick $N\in\mathbb{N}$ and $g\in G$. Let $n=n_{N}\in\mathbb{N}$ be the integer chosen before to obtain the subset $E_{n}\subset\widehat{A}_{n}$. We also pick $y\in hY_{n}$ with $h\in\Gamma$ and $z\in Z_{\tau}$ with $\tau\in X$, and set $w=(y,z)$. If $\tau\in p_{n}^{-1}(E_{n})$, then $(\phi_{g}\circ T_{N})w=g({}^{h}a_{h^{-1}p_{n}(\tau)})$ and

[TABLE]

where $\bar{g}$ denotes the image of $g$ in $\Gamma$. Thus $\phi_{g}\circ T_{N}=T_{N}\circ\phi_{g}$ at $w$. If $\tau\not\in p_{n}^{-1}(E_{n})$, then $(\phi_{g}\circ T_{N})w=g$, and $(T_{N}\circ\phi_{g})w=g$ because $g\tau\not\in p_{n}^{-1}(E_{n})$. Assertion (i) follows.

We prove assertion (ii). Let the group $\mathbb{T}$ act on $W$ by $t(y,z)=(y,tz)$ for $t\in\mathbb{T}$, $y\in Y$ and $z\in Z$. Since $\mathbb{T}$ is compact, the action $\mathbb{T}\curvearrowright W$ is isomorphic to the action $\mathbb{T}\curvearrowright D\times\mathbb{T}$ given by $t(w,s)=(w,ts)$ for $t,s\in\mathbb{T}$ and $w\in D$, where $D$ is a Borel subset of $W$ which is the product of $Y$ with a Borel fundamental domain for the action $\mathbb{T}\curvearrowright Z$.

We pick $N\in\mathbb{N}$ and let $n=n_{N}$. For $y\in gY_{n}$ with $g\in\Gamma$ and $z\in Z_{\tau}$ with $\tau\in X$, if $\tau$ belongs to $p_{n}^{-1}(E_{n})$, then

[TABLE]

and otherwise $T_{N}^{\circ}(y,z)=(y,z)$. This shows that for each $y\in Y$ and $\tau\in X$, the map $T_{N}^{\circ}$ preserves the set $\{y\}\times Z_{\tau}$, and on that set, the map $T_{N}^{\circ}$ is equal to the transformation given by some single element of $\mathbb{T}$. Moreover $\{y\}\times Z_{\tau}$ is $\mathbb{T}$-invariant. Therefore if $T_{N}^{\circ}$ is regarded as a automorphism of $D\times\mathbb{T}$ under the isomorphism between $W$ and $D\times\mathbb{T}$, then $T_{N}^{\circ}$ preserves each orbit $\{w\}\times\mathbb{T}$ with $w\in D$, and on that orbit, the map $T_{N}^{\circ}$ is equal to the transformation given by some single element of $\mathbb{T}$. By inequality (5.9), those elements of $\mathbb{T}$, i.e., the value $\langle g^{-1}\tau,a_{g^{-1}p_{n}(\tau)}\rangle$ in equation (5.10), are uniformly close to $1$ if $N$ is so large that $\exp(2\pi i/N)$ is close to $1$. Thus assertion (ii) follows.

We pick $N\in\mathbb{N}$ and let $n=n_{N}$. If $y\in gY_{n}$ with $g\in\Gamma$ and $z\in Z_{\tau}$ with $\tau\in p_{n}^{-1}(E_{n})$, then the value $\langle g^{-1}\tau,a_{g^{-1}p_{n}(\tau)}\rangle\in\mathbb{T}$ has order more than $N$ by inequality (5.9). Moreover freeness of the action $\mathbb{T}\curvearrowright Z$, shown in Proposition 5.1 (ii), and equation (5.10) imply that $(y,z)$ is a periodic point of $T_{N}^{\circ}$ whose period is more than $N$. Assertion (iii) follows from this together with the inequality $|E_{n}|/|\widehat{A}_{n}|\geq 1-1/N$. ∎

We are going to apply Theorem 2.5. Let us check that the assumption in it is fulfilled for the p.m.p. action $G\curvearrowright(W,\omega)$, the $G$-equivariant measure-preserving map $\pi\colon(W,\omega)\to(X,\mu)$ and the central sequence $(T_{N})$ in the full group $[G\ltimes(W,\omega)]$, where we define the map $\pi$ by $\pi(y,z)=\tau$ for $y\in Y$ and $z\in Z_{\tau}$ with $\tau\in X$. We first note that $(T_{N})$ is indeed central by Lemma 5.2 (i) and (ii). The stabilizer of a point of $W$ in $G$ depends only on its image under $\pi$. Indeed the action $\mathcal{G}_{\tilde{\sigma}}\curvearrowright(Z,\zeta)$ is essentially free by Proposition 5.1 (iv) and thus the stabilizer of almost every $w\in W$ is equal to the kernel of $\pi(w)\in X=\widehat{A}$. As pointed out in the proof of Lemma 5.2 (ii), $T_{N}^{\circ}$ preserves the set of the form $\{y\}\times Z_{\tau}$ with $y\in Y$ and $\tau\in X$ and thus preserves each fiber of $\pi$. For each $w\in W$, since $A$ is abelian and the kernel of $\pi(w)$ is a subgroup of $A$, the element $T_{N}w\in A$ belongs to the centralizer of the stabilizer of $w$ in $G$. The inequality $\omega(B_{N})\geq 1-1/N$ shown in Lemma 5.2 (iii) implies that $\omega(\{\,w\in W\mid T_{N}^{\circ}w\neq w\,\})\to 1$ as $N\to\infty$. By Lemma 5.2 (iii) again, for each $p\in\mathbb{N}$, if $B_{N}^{p}\subset W$ denotes the set of $p$-periodic points of $T_{N}^{\circ}$, then $\omega(B_{N}^{p})\to 0$ as $N\to\infty$. Thus the assumption in Theorem 2.5 is fulfilled, and by the theorem, $G$ has the Schmidt property.

5.4. The other case

Let $G$ be a countable group and let $A$ be an infinite abelian normal subgroup of $G$ contained in the FC-center of $G$. Suppose that each finite subset of $A$ has finite normal closure in $G$ and let $A_{1}<A_{2}<\cdots$ be a strictly increasing sequence of finite subgroups of $A$ such that each $A_{n}$ is normalized by $G$. In this subsection, we suppose that condition $(\star)$ in Subsection 3.2 does not hold for this sequence and then construct a free p.m.p. Schmidt action of $G$. By Lemma 3.4, we may assume without loss of generality that there exists a prime number $p$ such that each $A_{n}$ is isomorphic to the direct sum of finitely many copies of $\mathbb{Z}/p\mathbb{Z}$. We may also assume that $A=\bigcup_{n}A_{n}$ and that $G/A$ is infinite as in the previous subsection.

We set $\Gamma=G/A$ and choose a section $\mathsf{s}\colon\Gamma\to G$ of the quotient map with $\mathsf{s}(e)=e$. We then obtain the 2-cocycle $\sigma\colon\Gamma\times\Gamma\to A$. We define $X$ as the group of homomorphisms from $A$ into the direct product $L\coloneqq\prod_{\mathbb{N}}\mathbb{Z}/p\mathbb{Z}$, while $X$ denoted the dual group $\widehat{A}$ of $A$ in the previous subsection. Let $\mu$ be the normalized Haar measure on $X$. Note that if we fix an embedding of $\mathbb{Z}/p\mathbb{Z}$ into the torus $\mathbb{T}$, then the dual $\widehat{A}$ is identified with the group of homomorphisms from $A$ into $\mathbb{Z}/p\mathbb{Z}$ since all elements of $A=\bigcup_{n}A_{n}$ except for the trivial one have order $p$. Under this identification, we often identify $X$ with the product group $\prod_{\mathbb{N}}\widehat{A}$ unless there is cause of confusion.

We recall the construction in Subsection 5.1. Define the action of $G$ on $X$ by $(g\tau)(a)=\tau(g^{-1}ag)$ for $g\in G$, $a\in A$ and $\tau\in X$, which induces the action of $\Gamma$ on $X$. Let $\mathcal{U}=X\times L$ be the bundle over $X$, which is a groupoid with unit space $X$. Then we obtain the 2-cocycle $\tilde{\sigma}\colon(X\rtimes\Gamma)^{(2)}\to\mathcal{U}$ by formula (5.1) and obtain the groupoid extension

[TABLE]

together with the homomorphism $\eta\colon X\rtimes G\to\mathcal{G}_{\tilde{\sigma}}$ such that

[TABLE]

and $\eta(\tau,a)=(\tau,\tau(a))\in\mathcal{U}$ for all $a\in A$ and $\tau\in X$.

Lemma 5.3**.**

With the above notation,

1. (i)

for each $N\in\mathbb{N}$, the set of points $\tau=(\tau_{i})_{i\in\mathbb{N}}\in X$ such that $\bigcap_{i=1}^{N}\ker\tau_{i}=\ker\tau$ is $\mu$-null. 2. (ii)

For $\mu$-almost every $\tau\in X$, we have $\ker\tau=\{e\}$. Therefore the groupoid $X\rtimes A$ embeds into $\mathcal{U}$ via $\eta$ if it is restricted to some $\mu$-conull subset of $X$.

Proof.

The set in assertion (i) is written as

[TABLE]

We note that if $a$ is a non-trivial element of $A$, then the subgroup $\{\,\xi\in\widehat{A}\mid a\in\ker\xi\,\}$ is of index $p$ in $\widehat{A}$ and thus has measure $1/p$, where $\widehat{A}$ is equipped with the normalized Haar measure. Then for each $\tau_{1},\ldots,\tau_{N}\in\widehat{A}$, the set $\{\,\xi\in\widehat{A}\mid\bigcap_{i=1}^{N}\ker\tau_{i}<\ker\xi\,\}$ has measure at most $1/p$ because this is contained in the set $\{\,\xi\in\widehat{A}\mid a\in\ker\xi\,\}$ if $a$ is chosen to be a non-trivial element of $\bigcap_{i=1}^{N}\ker\tau_{i}$. By Fubini’s theorem, the set in (5.11) is $\mu$-null.

For each non-trivial $a\in A$, the set $\{\,\tau\in X\mid a\in\ker\tau\,\}$ is identified with the product set $\prod_{\mathbb{N}}\{\,\xi\in\widehat{A}\mid a\in\ker\xi\,\}$ and hence $\mu$-null. Assertion (ii) follows. ∎

Let $\mathcal{G}_{\tilde{\sigma}}\curvearrowright(Z,\zeta)$ be the free p.m.p. action constructed in Subsection 5.2, i.e., the action co-induced from the shift action of $\mathcal{U}$ on itself. The space $Z$ is fibered over $X$. The fiber at $\tau\in X$ is denoted by $Z_{\tau}$. For $n\in\mathbb{N}$, let $\Gamma_{n}$ be the group of elements of $\Gamma$ acting on $A_{n}$ trivially. Let $\Gamma\curvearrowright(Y,\nu)$ be the profinite p.m.p. action associated with the system of the p.m.p. action $\Gamma\curvearrowright\Gamma/\Gamma_{n}$ given by left multiplication. As with the previous subsection, let $\mathcal{G}_{\tilde{\sigma}}$ act on $Y\times Z$ diagonally, where $Y\times Z$ is fibered over $X$ with respect to the map sending each element of $Y\times Z_{\tau}$ to $\tau$ for each $\tau\in X$. Through the homomorphism $\eta\colon X\rtimes G\to\mathcal{G}_{\tilde{\sigma}}$, we obtain the p.m.p. action of $G$ on the product space $(W,\omega)\coloneqq(Y\times Z,\nu\times\zeta)$. We note that the action $G\curvearrowright(W,\omega)$ is essentially free because the action $\mathcal{G}_{\tilde{\sigma}}\curvearrowright(Z,\zeta)$ is essentially free by Proposition 5.1 (iv) and $\ker\eta$ is trivial in the sense of Lemma 5.3 (ii).

We now construct a central sequence $(T_{N})$ in the full group of the translation groupoid $G\ltimes(W,\omega)$. Pick $N\in\mathbb{N}$. For each $a\in A$, we set

[TABLE]

By Lemma 5.3 (i), $X=\bigcup_{a\in A}X_{a}$ up to null sets. Let $Y_{n}$ denote the inverse image of the coset $e\Gamma_{n}$ under the projection from $Y$ onto $\Gamma/\Gamma_{n}$. Then

[TABLE]

where we set $A_{0}=\{e\}$. If $a\in A_{n}\setminus A_{n-1}$ and $g,h\in\Gamma$, then $h(X_{a}\times gY_{n})=X_{h\cdot a}\times hgY_{n}$ with respect to the diagonal action $\Gamma\curvearrowright X\times Y$, where the dot stands for the action of $\Gamma$ on $A$. Thus the saturation $\Gamma(X_{a}\times gY_{n})$ is the disjoint union of the translates $h(X_{a}\times gY_{n})$ with $h$ running through representatives of elements of $\Gamma/\Gamma_{n}$. Let us call such a subset a $(\Gamma/\Gamma_{n})$-base, that is, call a Borel subset $B\subset X\times Y$ a $(\Gamma/\Gamma_{n})$-base if $B$ is $\Gamma_{n}$-invariant and the saturation $\Gamma B$ is the disjoint union of the translates $hB$ with $h$ running through representatives of elements of $\Gamma/\Gamma_{n}$.

Lemma 5.4**.**

With the above notation, there exist Borel subsets of $X$, $B_{1},B_{2},\ldots$, such that $X\times Y=\bigsqcup_{m=1}^{\infty}\Gamma B_{m}$ and each $B_{m}$ is a $(\Gamma/\Gamma_{n})$-base contained in $X_{a}\times gY_{n}$ for some $n\in\mathbb{N}$, $a\in A_{n}\setminus A_{n-1}$ and $g\in\Gamma$.

Proof.

For each $n\in\mathbb{N}$, let $D(n,1),D(n,2),\ldots,D(n,k_{n})$ be an enumeration of the $(\Gamma/\Gamma_{n})$-bases $X_{a}\times gY_{n}$ indexed by $a\in A_{n}\setminus A_{n-1}$ and a representative $g$ of an element of $\Gamma/\Gamma_{n}$, with $k_{n}=|A_{n}\setminus A_{n-1}|\,|\Gamma/\Gamma_{n}|$. Let $(E_{m})_{m\in\mathbb{N}}$ be the enumeration of the sets $D(n,k)$ with respect to the lexicographic order of the indices $(n,k)$.

We inductively define a Borel subset $B_{m}\subset X\times Y$. We set $B_{1}=E_{1}$. Suppose that $B_{1},\ldots,B_{m-1}$ are defined. We set $B_{m}=E_{m}\setminus\bigcup_{i=1}^{m-1}\Gamma B_{i}$. Then $E_{m}=D(n,k)$ for some $n$ and $k$ and thus $B_{m}$ is a $(\Gamma/\Gamma_{n})$-base. By construction $\Gamma B_{m}$ and $\Gamma B_{l}$ are disjoint for all distinct $m$, $l$. Since the sets $E_{m}$ cover $X\times Y$, the sets $\Gamma B_{m}$ cover $X\times Y$. ∎

We define a map $T_{N}\colon W\to A$ as follows: Let $q\colon W\to X\times Y$ be the projection that sends a point $(y,z)\in W$ with $z\in Z_{\tau}$ and $\tau\in X$ to the point $(\tau,y)$. By Lemma 5.4, the set $X\times Y$ is covered by the mutually disjoint sets $\Gamma B_{m}$ with $m\in\mathbb{N}$. For each $m\in\mathbb{N}$, we have $n_{m}\in\mathbb{N}$, $a_{m}\in A_{n_{m}}\setminus A_{n_{m}-1}$ and $g_{m}\in\Gamma$ such that the set $B_{m}$ is a $(\Gamma/\Gamma_{n_{m}})$-base contained in $X_{a_{m}}\times g_{m}Y_{n_{m}}$. For $w\in q^{-1}(hB_{m})$ with $h\in\Gamma$, we set

[TABLE]

This is well-defined because $B_{m}$ is a $(\Gamma/\Gamma_{n_{m}})$-base and $a_{m}$ is fixed by $\Gamma_{n_{m}}$. The map from $W$ into itself, $w\mapsto(T_{N}w)w$, is an automorphism of $W$ because $A$ preserves each fiber of $q$. Thus $T_{N}$ is an element of the full group $[G\ltimes(W,\omega)]$.

Lemma 5.5**.**

With the above notation,

1. (i)

for every $N\in\mathbb{N}$ and $g\in G$, we have $\phi_{g}\circ T_{N}=T_{N}\circ\phi_{g}$, where $\phi_{g}\colon X\to G$ is the element of the full group $[G\ltimes(W,\omega)]$ given by the constant map with value $g$. 2. (ii)

For every Borel subset $B\subset W$, we have $\omega(T_{N}^{\circ}B\bigtriangleup B)\to 0$ as $N\to\infty$. 3. (iii)

For every $N\in\mathbb{N}$ and every $w\in W$, we have $T_{N}^{\circ}w\neq w$.

Proof.

We prove assertion (i). If $w\in q^{-1}(hB_{m})$ with $h\in\Gamma$, then we have $(T_{N}\circ\phi_{g})w=T_{N}(gw)g=((\bar{g}h)\cdot a_{m})g$ with $\bar{g}$ the image of $g$ in $\Gamma$, and also have $(\phi_{g}\circ T_{N})w=g(h\cdot a_{m})$. These two coincide.

We prove assertion (ii). The proof is similar to that of Lemma 5.2 (ii). Using the action of $\mathcal{U}$ on $Z$, which restricts the action of $\mathcal{G}_{\tilde{\sigma}}$, we define an action of $L$ on $Z$ by $lf=(\tau,l)f$ for $l\in L$ and $f\in Z_{\tau}$ with $\tau\in X$. This is the action defined in Proposition 5.1 (ii). Let $L$ act on $W$ by $l(y,z)=(y,lz)$ for $l\in L$, $y\in Y$ and $z\in Z$.

Fix $N\in\mathbb{N}$. Recall that the group $A$ acts on $W$ via the homomorphism $\eta\colon X\rtimes G\to\mathcal{G}_{\tilde{\sigma}}$, which satisfies $\eta(\tau,a)=(\tau,\tau(a))$ for all $\tau\in X$ and $a\in A$. Hence if $w=(y,z)\in q^{-1}(hB_{m})$ with $z\in Z_{\tau}$, $\tau=(\tau_{i})_{i\in\mathbb{N}}\in X$ and $h\in\Gamma$, then

[TABLE]

Since $q(w)=(\tau,y)\in hB_{m}$, we have $\tau\in X_{h\cdot a_{m}}$ and thus $\tau_{1}(h\cdot a_{m})=\cdots=\tau_{N}(h\cdot a_{m})=0$ and $\tau(h\cdot a_{m})\neq 0$. This says that the element $\tau(h\cdot a_{m})\in L=\prod_{\mathbb{N}}\mathbb{Z}/p\mathbb{Z}$ is non-trivial and is close to the identity if $N$ is large. The definition of $T_{N}w$ depends only on $q(w)$, and the action of $L$ on $W$ preserves each fiber of $q$. Hence on each $L$-orbit in $W$, the map $T_{N}^{\circ}$ is equal to the transformation given by some single element of $L$. Assertion (ii) then follows from the existence of a Borel fundamental domain for the action $L\curvearrowright Z$ as well as in the proof of Lemma 5.2 (ii).

Assertion (iii) follows from the condition $\tau(h\cdot a_{m})\neq 0$ shown above and freeness of the action of $L$ on $Z$ shown in Proposition 5.1 (ii). ∎

Therefore the groupoid $G\ltimes(W,\omega)$ is Schmidt, and so is its almost every ergodic component by Lemma 2.2. We have already shown that the action $G\curvearrowright(W,\omega)$ is essentially free, in the paragraph after Lemma 5.3. Thus $G$ has the Schmidt property.

6. Another construction using ultraproducts

Let $G$ be a countable group with infinite FC-center. We construct a free p.m.p. Schmidt action of $G$ by way of ultraproducts. This construction is self-contained and independent of the construction given so far.

Step 1. Setting up the sequence of actions: Let $A$ denote the FC-center of $G$. Then $A$ has an infinite abelian subgroup $B$, which is found as follows: First, pick a non-trivial $a_{1}\in A$. If $\langle a_{1}\rangle$ is infinite, let $B=\langle a_{1}\rangle$. Otherwise pick an element $a_{2}$ of the set $C_{A}(a_{1})\setminus\langle a_{1}\rangle$, which is non-empty because $C_{A}(a_{1})$ is of finite index in $A$ and hence infinite. If $\langle a_{1},a_{2}\rangle$ is infinite, let $B=\langle a_{1},a_{2}\rangle$. Otherwise pick an element $a_{3}$ of the set $C_{A}(a_{1},a_{2})\setminus\langle a_{1},a_{2}\rangle$, which is non-empty by the same reason. Repeat this procedure. Then either it stops in finite steps and the group $B=\langle a_{1},\ldots,a_{n}\rangle$ for some $n$ is infinite and abelian, or it does not stop and the group $B=\langle a_{1},a_{2},\ldots\rangle$ is infinite and abelian.

We may write $B$ as an increasing union of finitely generated subgroups $B=\bigcup_{n\in\mathbb{N}}B_{n}$. Let $G_{n}\coloneqq C_{G}(B_{n})$, so that $G_{n}$ is a finite index subgroup of $G$ which contains $B$. Since $B$ is abelian, we may find a free ergodic compact action $B\curvearrowright^{\beta}(Y,\mu_{Y})$ of $B$, where $Y$ is a compact abelian metrizable group and $\beta\colon B\rightarrow Y$ is an injective homomorphism with dense image, and $B$ is acting on $Y$ by left translation via $\beta$. Let $G_{n}\curvearrowright^{\beta_{n}}(Y,\mu_{Y})^{G_{n}/B}$ be the p.m.p. action co-induced from the action $\beta$ of $B$. Explicitly, this is defined as follows: We pick a section $t_{n}\colon G_{n}/B\rightarrow G_{n}$ of the projection map $G_{n}\rightarrow G_{n}/B$ with $t_{n}(eB)=e$, and we let $w_{n}\colon G_{n}\times G_{n}/B\rightarrow B$ be the associated cocycle for the action $G_{n}\curvearrowright G_{n}/B$ given by $w_{n}(g,hB)\coloneqq t_{n}(ghB)^{-1}gt_{n}(hB)$ for $g,h\in G_{n}$. Then the action $G_{n}\curvearrowright^{\beta_{n}}Y^{G_{n}/B}$ is given by

[TABLE]

for $g,h\in G_{n}$. For each $n$, pick a section $s_{n}\colon G/G_{n}\rightarrow G$ of the projection map $G\rightarrow G/G_{n}$ with $s_{n}(eG_{n})=e$, and let $v_{n}\colon G\times G/G_{n}\rightarrow G_{n}$ be the associated cocycle for the p.m.p. action $G\curvearrowright(G/G_{n},\mu_{G/G_{n}})$ (where $\mu_{G/G_{n}}$ is the normalized counting measure), given by $v_{n}(g,hG_{n})\coloneqq s_{n}(ghG_{n})^{-1}gs_{n}(hG_{n})$ for $g,h\in G$. Then we equip $Z_{n}\coloneqq G/G_{n}\times Y^{G_{n}/B}$ with the product measure $\eta_{n}\coloneqq\mu_{G/G_{n}}\times\mu_{Y}^{G_{n}/B}$ and we let $G\curvearrowright^{\alpha_{n}}(Z_{n},\eta_{n})$ be the skew product action, which is the p.m.p. action defined by

[TABLE]

for $g\in G$ and $(kG_{n},x)\in Z_{n}$.

Step 2. The ultraproduct and its quotients: Fix a non-principal ultrafilter $\mathcal{V}$ on $\mathbb{N}$ and let $G\curvearrowright^{\alpha}(Z_{\mathcal{V}},\eta_{\mathcal{V}})$ be the ultraproduct of the sequence of actions $(G\curvearrowright^{\alpha_{n}}(Z_{n},\eta_{n}))_{n\in\mathbb{N}}$ with respect to $\mathcal{V}$. Thus $Z_{\mathcal{V}}=(\prod_{n}Z_{n})/\!\sim_{\mathcal{V}}$, where $\sim_{\mathcal{V}}$ is the equivalence relation on $\prod_{n}Z_{n}$ such that $(y_{n})\sim_{\mathcal{V}}(z_{n})$ if and only if $\{\,n\in\mathbb{N}\mid y_{n}=z_{n}\,\}\in\mathcal{V}$; we write $[(z_{n})]_{\mathcal{V}}$ for the equivalence class of the sequence $(z_{n})$. For a sequence $(D_{n})$ of Borel sets $D_{n}\subset Z_{n}$, let $[(D_{n})]_{\mathcal{V}}$ be the associated basic measurable subset of $Z_{\mathcal{V}}$, i.e.,

[TABLE]

where $1_{D_{n}}$ is the indicator function of $D_{n}$. The assignment $[(D_{n})]_{\mathcal{V}}\mapsto\lim_{n\rightarrow\mathcal{V}}\eta_{n}(D_{n})$ defines a premeasure on the algebra of all such basic measurable sets, and hence this assignment extends uniquely to a countably additive measure $\eta_{\mathcal{V}}$ on the completion $\mathcal{B}_{\mathcal{V}}$ of the sigma algebra generated by the basic measurable sets. This is how the measure $\eta_{\mathcal{V}}$ is defined. The action $\alpha$, of $G$ on $Z_{\mathcal{V}}$, is given by $\alpha(g)[(z_{n})]_{\mathcal{V}}\coloneqq[(gz_{n})]_{\mathcal{V}}$.

Likewise, let $G\curvearrowright(X_{\mathcal{V}},\mu_{\mathcal{V}})$ denote the ultraproduct, with respect to $\mathcal{V}$, of the sequence of actions $(G\curvearrowright(G/G_{n},\mu_{G/G_{n}}))_{n\in\mathbb{N}}$. Then the projection map $p\colon(Z_{\mathcal{V}},\eta_{\mathcal{V}})\rightarrow(X_{\mathcal{V}},\mu_{\mathcal{V}})$, $[(k_{n}G_{n},x_{n})]_{\mathcal{V}}\mapsto[(k_{n}G_{n})]_{\mathcal{V}}$, is measure-preserving and $G$-equivariant.

Let $G\curvearrowright(P,\mu_{P})$ denote the profinite action that is the inverse limit of the finite actions $G\curvearrowright G/G_{n}$. Elements of $P$ consist of sequences $(g_{m}G_{m})$ with $g_{m}G_{m}\supset g_{m+1}G_{m+1}$ for all $m$. For each $[(k_{n}G_{n})]_{\mathcal{V}}\in X_{\mathcal{V}}$ and each $m\in\mathbb{N}$, let $\Phi_{m}[(k_{n}G_{n})]_{\mathcal{V}}$ be the unique left coset $gG_{m}$ of $G_{m}$ for which the set $\{\,n\in\mathbb{N}\mid k_{n}G_{n}\subset gG_{m}\,\}$ belongs to $\mathcal{V}$. Then each $\Phi_{m}\colon X_{\mathcal{V}}\rightarrow G/G_{m}$ is $G$-equivariant and measure-preserving, and $\Phi_{m}[(k_{n}G_{n})]_{\mathcal{V}}\supset\Phi_{m+1}[(k_{n}G_{n})]_{\mathcal{V}}$, so we obtain the measure-preserving $G$-equivariant map $\Phi\colon(X_{\mathcal{V}},\mu_{\mathcal{V}})\rightarrow(P,\mu_{P})$ given by $\Phi[(k_{n}G_{n})]_{\mathcal{V}}=(\Phi_{m}[(k_{n}G_{n})]_{\mathcal{V}})_{m}$.

For each $n$, let $\pi_{n}\colon Z_{n}\rightarrow Y$ be the map $\pi_{n}(kG_{n},x)\coloneqq x(eB)$ projecting to the identity-coset coordinate of $x\in Y^{G_{n}/B}$. Let $\pi\colon Z_{\mathcal{V}}\rightarrow Y$ be defined by

[TABLE]

(note that this limit exists since $Y$ is compact). By [BTD, Proposition 8.4], this map is measurable and measure-preserving, with $\eta_{\mathcal{V}}(\pi^{-1}(E)\bigtriangleup[(\pi_{n}^{-1}(E))]_{\mathcal{V}})=0$ for every Borel subset $E$ of $Y$. Let $\mathcal{Y}$ denote the subalgebra of $\mathcal{B}_{\mathcal{V}}$ consisting of all sets of the form $\pi^{-1}(E)$ with $E\subset Y$ Borel, and let $\mathcal{P}$ denote the subalgebra of $\mathcal{B}_{\mathcal{V}}$ consisting of all sets of the form $(\Phi\circ p)^{-1}(C)$ with $C\subset P$ Borel.

Step 3. The central sequence: For each $b\in B$, the conjugacy class $b^{G}$ of $b$ in $G$ is finite, and the map $T_{b}\colon Z_{\mathcal{V}}\rightarrow b^{G}$ given by

[TABLE]

is well-defined, since if $m(b)\in\mathbb{N}$ is the least such that $G_{m(b)}<C_{G}(b)$ then for all $n\geq m(b)$ the conjugate $k_{n}bk_{n}^{-1}$ depends only on the coset $k_{n}G_{n}$ of $G_{n}$. Letting $(g_{m}G_{m})_{m\in\mathbb{N}}\coloneqq\Phi[(k_{n}G_{n})]_{\mathcal{V}}$, we have $\{\,n\in\mathbb{N}\mid k_{n}G_{n}\subset g_{m(b)}G_{m(b)}\,\}\in\mathcal{V}$ and hence $T_{b}[(k_{n}G_{n},x_{n})]_{\mathcal{V}}=g_{m(b)}bg_{m(b)}^{-1}=\lim_{m\rightarrow\infty}g_{m}bg_{m}^{-1}$. In particular, the map $T_{b}$ is $\mathcal{P}$-measurable. We have $T_{b}(gz)=gT_{b}(z)g^{-1}$ for all $g\in G$ and $z\in Z_{\mathcal{V}}$. The map $T_{b}^{\circ}\colon Z_{\mathcal{V}}\rightarrow Z_{\mathcal{V}}$ given by $T_{b}^{\circ}(z)=\alpha(T_{b}(z))z$ is an automorphism of $(Z_{\mathcal{V}},\eta_{\mathcal{V}})$ which commutes with $\alpha(g)$ for all $g\in G$. Then the map $p$ is $T_{b}^{\circ}$-invariant, and in particular every set in $\mathcal{P}$ is $T_{b}^{\circ}$-invariant.

For each $b\in B$ and $[(k_{n}G_{n},x_{n})]_{\mathcal{V}}\in Z_{\mathcal{V}}$, since the set $\{\,n\in\mathbb{N}\mid T_{b}[(k_{n}G_{n},x_{n})]_{\mathcal{V}}=k_{n}bk_{n}^{-1}\,\}$ belongs to $\mathcal{V}$, the transformation $T_{b}^{\circ}$ is given by

[TABLE]

For all large enough $n$, we have $G_{n}<C_{G}(b)$, and for such $n$, since $B<G_{n}$, we have $v_{n}(k_{n}bk_{n}^{-1},k_{n}G_{n})=v_{n}(k_{n},eG_{n})v_{n}(b,eG_{n})v_{n}(k_{n},eG_{n})^{-1}=b$. Since this holds for all large $n$, we obtain

[TABLE]

Also, for all $n$ with $G_{n}<C_{G}(b)$, for each $hB\in G_{n}/B$ we have $b^{-1}hB=hB$ and $w_{n}(b,b^{-1}hB)=b$, so that $(\beta_{n}(b)x_{n})(hB)=\beta(b)x_{n}(hB)$, and therefore

[TABLE]

Let $(b_{i})_{i\in\mathbb{N}}$ be a sequence of distinct elements in $B$ with $\beta(b_{i})$ converging weakly to the identity element of $Y$. Then for each Borel subset $E$ of $Y$, we have $\mu_{Y}(\beta(b_{i})E\bigtriangleup E)\rightarrow 0$ as $i\rightarrow\infty$, so it follows from (6.1) that $\eta_{\mathcal{V}}(T_{b_{i}}^{\circ}(\pi^{-1}(E))\bigtriangleup\pi^{-1}(E))\rightarrow 0$ as $i\rightarrow\infty$.

Thus both $\mathcal{P}$ and $\mathcal{Y}$ belong to the sigma subalgebra $\mathcal{D}$ of $\mathcal{B}_{\mathcal{V}}$ consisting of all $D\in\mathcal{B}_{\mathcal{V}}$ such that $\lim_{i\rightarrow\infty}\eta_{\mathcal{V}}(T_{b_{i}}^{\circ}D\bigtriangleup D)=0$. Since each $T_{b_{i}}$ commutes with $\alpha(G)$, the sigma algebra $\mathcal{D}$ is $\alpha(G)$-invariant.

Step 4. Ensuring essential freeness for the action of $A$ on the upcoming separable quotient: We pick $a\in A\setminus\{e\}$ and let $F_{a}\subset X_{\mathcal{V}}$ be the fixed point set of $a$ in $X_{\mathcal{V}}$. Then we have $X_{\mathcal{V}}\setminus F_{a}=[(C_{a,n})_{n}]_{\mathcal{V}}$, where $C_{a,n}\coloneqq\{\,kG_{n}\in G/G_{n}\mid akG_{n}\neq kG_{n}\,\}$. We can write the set $C_{a,n}$ as a union of three pairwise disjoint sets $C_{a,n,0}$, $C_{a,n,1}$, $C_{a,n,2}$ such that $aC_{a,n,i}\cap C_{a,n,i}=\emptyset$ (indeed let $C_{a,n,0}$ be a maximal subset of $C_{a,n}$ such that $aC_{a,n,0}\cap C_{a,n,0}=\emptyset$ and set $C_{a,n,1}\coloneqq aC_{a,n,0}\cap C_{a,n}$ and $C_{a,n,2}\coloneqq C_{a,n}\setminus(C_{a,n,0}\cup C_{a,n,1})$). Each of the sets $D_{a,i}\coloneqq(\Phi\circ p)^{-1}([(C_{a,n,i})_{n}]_{\mathcal{V}})$ is $T_{b}^{\circ}$-invariant for all $b\in B$ and hence belongs to $\mathcal{D}$. For $c\in a^{G}$, we define $F_{a,c}$ as the set of all $[(k_{n}G_{n})]_{\mathcal{V}}\in F_{a}$ for which $\lim_{n\rightarrow\mathcal{V}}s_{n}(k_{n}G_{n})^{-1}as_{n}(k_{n}G_{n})=c$, so that $F_{a,c}$ is a basic measurable subset of $X_{\mathcal{V}}$ corresponding to the sequence of sets $\{\,kG_{n}\in G/G_{n}\mid s_{n}(kG_{n})as_{n}(kG_{n})^{-1}=c\,\}$ with $n\in\mathbb{N}$. The sets $F_{a,c}$ with $c\in a^{G}$ partition $F_{a}$. Each of the sets $p^{-1}(F_{a,c})$ is $T_{b}^{\circ}$-invariant for all $b\in B$ and hence belongs to $\mathcal{D}$.

Step 5. Defining the separable quotient of the ultraproduct: Since $\mathcal{D}$ is $G$-invariant and both the algebras $\mathcal{P}$ and $\mathcal{Y}$ are countably generated and $G$ is countable, we can find a countably generated $G$-invariant sigma subalgebra $\mathcal{D}_{0}$ of $\mathcal{D}$ which contains both $\mathcal{P}$ and $\mathcal{Y}$ as well as all of the sets $D_{a,i}$ and $p^{-1}(F_{a,c})$ for $a\in A\setminus\{e\}$, $c\in a^{G}$ and $i\in\{0,1,2\}$. Then we may find a point realization $G\curvearrowright(W_{0},\mu_{0})$ for the action of $G$ on the measure algebra $\mathcal{D}_{0}$, along with a $G$-equivariant measure-preserving map $\varphi\colon(Z_{\mathcal{V}},\eta_{\mathcal{V}})\rightarrow(W_{0},\mu_{0})$ which is a point realization of the measure algebra inclusion $\mathcal{D}_{0}\hookrightarrow\mathcal{B}_{\mathcal{V}}$. For each $b\in B$, since the map $T_{b}$ is $\mathcal{P}$-measurable and $\mathcal{P}\subset\mathcal{D}_{0}$, $T_{b}$ descends via $\varphi$ to a map $S_{b}\colon W_{0}\rightarrow b^{G}$, which satisfies $S_{b}(gw)=gS_{b}(w)g^{-1}$ for all $g\in G$ and $w\in W_{0}$. The map $S_{b}^{\circ}\colon W_{0}\rightarrow W_{0}$ given by $S_{b}^{\circ}(w)=S_{b}(w)w$ is an automorphism of $(W_{0},\mu_{0})$ with $\varphi\circ T_{b}^{\circ}=S_{b}^{\circ}\circ\varphi$. Since $\mathcal{Y}\subset\mathcal{D}_{0}$ is invariant under the group $\{\,T_{b}^{\circ}\mid b\in B\,\}$, the map $\pi$ descends to a measure-preserving map $\pi_{0}\colon(W_{0},\mu_{0})\rightarrow(Y,\mu_{Y})$ with $\pi_{0}(S_{b}^{\circ}w)=\beta(b)\pi_{0}(w)$ for all $b\in B$. It follows that the group $\{\,S_{b}^{\circ}\mid b\in B\,\}$ acts essentially freely on $W_{0}$ since $\beta(B)$ acts freely on $Y$.

Since $\mathcal{D}_{0}\subset\mathcal{D}$, it follows that $(S_{b_{i}})_{i\in\mathbb{N}}$ is a central sequence in the full group of the action $G\curvearrowright(W_{0},\mu_{0})$ with $S_{b_{i}}^{\circ}w\neq w$ for almost every $w\in W_{0}$. However, it is not clear whether this action of $G$ is essentially free, so we take an essentially free action $G/A\curvearrowright(W_{1},\mu_{1})$ and let $G\curvearrowright(W_{0}\times W_{1},\mu_{0}\times\mu_{1})$ be the product action, where $G$ acts on $W_{1}$ via the projection onto $G/A$. Then each $S_{b}\colon W_{0}\to b^{G}$ lifts to the map $\tilde{S}_{b}\colon W_{0}\times W_{1}\to b^{G}$ via the projection from $W_{0}\times W_{1}$ onto $W_{0}$, and it satisfies $\tilde{S}_{b}(gw)=g\tilde{S}_{b}(w)g^{-1}$ for all $g\in G$ and $w\in W_{0}\times W_{1}$. The map $\tilde{S}_{b}^{\circ}$ is given by $\tilde{S}_{b}^{\circ}(w_{0},w_{1})=S_{b}(w_{0})(w_{0},w_{1})=(S_{b}^{\circ}(w_{0}),w_{1})$ and hence an automorphism of $W_{0}\times W_{1}$, and the group $\{\,\tilde{S}_{b}^{\circ}\mid b\in B\,\}$ acts essentially freely on $W_{0}\times W_{1}$. Since $A$ acts trivially on $W_{1}$, it follows that $(\tilde{S}_{b_{i}})_{i\in\mathbb{N}}$ is a central sequence in the full group of the action $G\curvearrowright(W_{0}\times W_{1},\mu_{0}\times\mu_{1})$, and it satisfies $\tilde{S}_{b_{i}}^{\circ}w\neq w$ for almost every $w\in W_{0}\times W_{1}$.

Thus we will be done once we show that the action $G\curvearrowright(W_{0}\times W_{1},\mu_{0}\times\mu_{1})$ is essentially free. For this, it is enough to show that the action $A\curvearrowright(W_{0},\mu_{0})$ is essentially free.

Step 6. Verifying that the action $A\curvearrowright(W_{0},\mu_{0})$ is essentially free: Fix $a\in A\setminus\{e\}$. Suppose that there is some $c\in a^{G}$ for which the set $F_{a,c}$ has positive measure. We first show that for almost every $z\in p^{-1}(F_{a,c})$, $\pi(\alpha(a)z)$ and $\pi(z)$ are distinct. Since $F_{a,c}$ is a subset of $F_{a}$, if $[(k_{n}G_{n})]_{\mathcal{V}}\in F_{a,c}$ then for $\mathcal{V}$-almost every $n\in\mathbb{N}$, we have $v_{n}(a,k_{n}G_{n})=s_{n}(k_{n}G_{n})^{-1}as_{n}(k_{n}G_{n})=c$ and hence $c\in G_{n}$. Since the sequence $(G_{n})$ is decreasing, this implies $c\in G_{n}$ for all $n\in\mathbb{N}$, and hence the element $\beta(w_{n}(c,c^{-1}B))\in Y$ is well-defined for all $n$. Let $y_{c}$ denote the limit along $\mathcal{V}$ of this sequence, $y_{c}\coloneqq\lim_{n\rightarrow\mathcal{V}}\beta(w_{n}(c,c^{-1}B))\in Y$. For each $z=[(k_{n}G_{n},x_{n})]_{\mathcal{V}}\in p^{-1}(F_{a,c})$, we have $\alpha(a)z=[(k_{n}G_{n},\beta_{n}(c)x_{n})]_{\mathcal{V}}$, and hence

[TABLE]

To see these are almost surely distinct, we consider the two possibilities of whether $c\in B$ or $c\not\in B$. If $c\in B$ then $x_{n}(c^{-1}B)=x_{n}(eB)$ and $y_{c}=\lim_{n\rightarrow\mathcal{V}}\beta(w_{n}(c,B))=\beta(c)\neq e$, and hence $\pi(\alpha(a)z)=\beta(c)\pi(z)\neq\pi(z)$, as was to be shown. Suppose now that $c\not\in B$. By [BTD, Proposition 8.4], the map $\pi_{c}\colon(Z_{\mathcal{V}},\eta_{\mathcal{V}})\rightarrow(Y,\mu_{Y})$ defined by $\pi_{c}[(k_{n}G_{n},x_{n})]_{\mathcal{V}}\coloneqq y_{c}\lim_{n\rightarrow\mathcal{V}}x_{n}(c^{-1}B)$ is measurable and measure-preserving, and for each Borel subset $E$ of $Y$, we have $\eta_{\mathcal{V}}(\pi_{c}^{-1}(E)\bigtriangleup[(\pi_{c,n}^{-1}(E))_{n}]_{\mathcal{V}})=0$, where the map $\pi_{c,n}\colon(Z_{n},\eta_{n})\rightarrow(Y,\mu_{Y})$ is defined by $\pi_{c,n}(kG_{n},x)\coloneqq y_{c}x(c^{-1}B)$. Since $c\not\in B$, the random variables $\pi_{n}$, $\pi_{c,n}$ are independent for every $n$. Therefore the random variables $\pi$, $\pi_{c}$ are also independent. Since $\mu_{Y}$ is atomless, it follows that $\pi(z)\neq\pi_{c}(z)$ for almost every $z\in Z_{\mathcal{V}}$. By (6.2), for almost every $z\in p^{-1}(F_{a,c})$, we thus have $\pi(\alpha(a)z)=\pi_{c}(z)\neq\pi(z)$, as was to be shown.

It now follows that $\pi(\alpha(a)z)\neq\pi(z)$ for almost every $z\in p^{-1}(F_{a})$. Since $\pi=\pi_{0}\circ\varphi$ and since each of the sets $p^{-1}(F_{a})$ belongs to $\mathcal{D}_{0}$, it follows that $\pi_{0}(aw)\neq\pi_{0}(w)$ and hence $aw\neq w$ for almost every $w\in\varphi(p^{-1}(F_{a}))$. In addition, since each of the sets $D_{a,i}$ for $i\in\{0,1,2\}$ belongs to $\mathcal{D}_{0}$, it follows that $aw\neq w$ for almost every $w\in W_{0}\setminus\varphi(p^{-1}(F_{a}))$. This shows that the action of $A$ on $W_{0}$ is essentially free.

Appendix A A Kazhdan group with prescribed center

Given a countable abelian group $A$, we construct a countable group $G$ with property (T) such that the center of $G$ is isomorphic to $A$. We rely on the construction of Cornulier [C] as well as in Examples 3.6 and 3.7. Let $R\coloneqq\mathbb{Z}[t]$ be the ring of polynomials over $\mathbb{Z}$ in one indeterminate $t$. In the course of the construction, we will use property (T) of the group $\mathit{SL}_{3}(R)$ (e.g., [EJZ, Theorem 1.1] and [M, Theorem 1.8]) and property (T) of the pair $(\mathit{SL}_{3}(R)\ltimes R^{3},R^{3})$ ([Ka, Theorem 1.9, a)]). Note that the statements in those papers are given in terms of the group generated by elementary matrices in $\mathit{SL}_{3}(R)$, which is in fact equal to $\mathit{SL}_{3}(R)$ by [Su, Corollary 6.6].

Let $H$ be the subgroup of $\mathit{SL}_{5}(R)$ consisting of matrices of the form

[TABLE]

where $h\in\mathit{SL}_{3}(R)$, $c\in R$, and $u$ and $v$ are row and column vectors of $R^{3}$, respectively. Let $C$ be the center of $H$, which consists of matrices $g$ such that $h=I$, $u=0$ and $v=0$. Then $H/C$ is isomorphic to the semi-direct product $\Gamma\coloneqq\mathit{SL}_{3}(R)\ltimes(R^{3}\times R^{3})$, where $\mathit{SL}_{3}(R)$ acts on $R^{3}\times R^{3}$ by $h(u,v)=(uh^{-1},hv)$ for $h\in\mathit{SL}_{3}(R)$, a row vector $u\in R^{3}$, and a column vector $v\in R^{3}$. In fact, the map sending a matrix $g\in H$ of the form (A.1) to the element $(h,(u,h^{-1}v))$ of $\Gamma$ induces an isomorphism.

The group $\Gamma$ has property (T). To see this, recall the following fact: If $G$ is a countable group and $N$ is a normal subgroup of $G$ such that the group $G/N$ and the pair $(G,N)$ have property (T), then $G$ has property (T) ([BHV, Remark 1.7.7]). Property (T) of the group $\mathit{SL}_{3}(R)$ and the pair $(\mathit{SL}_{3}(R)\ltimes R^{3},R^{3})$ thus implies that $\mathit{SL}_{3}(R)\ltimes R^{3}$ has property (T). The group $\Gamma$ is written as the semi-direct product $(\mathit{SL}_{3}(R)\ltimes R^{3})\ltimes R^{3}$, and the above fact again implies that $\Gamma$ has property (T).

Hence the group $H/C$ has property (T). The commutator subgroup $[H,H]$ contains $C$, and thus the abelianization $H/[H,H]$ is finite. It follows from [BHV, Theorem 1.7.11] that $H$ has property (T).

We obtained the group $H$ with property (T) whose center $C$ is isomorphic to $R$ and to the direct sum $\bigoplus_{\mathbb{N}}\mathbb{Z}$. Let $A$ be an arbitrary countable abelian group. There exists a surjection from $C$ onto $A$. Let $C_{1}$ be the kernel of this surjection and set $H_{1}=H/C_{1}$. The group $H_{1}$ has property (T) and has the central subgroup $C/C_{1}$ isomorphic to $A$. In fact, the center of $H_{1}$ is exactly $C/C_{1}$ because $H/C\simeq\Gamma$ has trivial center.