On Benjamini--Schramm limits of congruence subgroups

TL;DR

This paper proves that sequences of orbifolds derived from non-conjugate congruence lattices in higher rank semisimple groups over local fields of zero characteristic converge in the Benjamini--Schramm sense to their universal cover.

Contribution

It establishes Benjamini--Schramm convergence for sequences of orbifolds associated with non-conjugate congruence lattices in higher rank semisimple groups.

Findings

Sequences of orbifolds from non-conjugate congruence lattices are Benjamini--Schramm convergent.

Convergence occurs to the universal cover in higher rank semisimple groups.

Results apply to local fields of zero characteristic.

Abstract

Every sequence of orbifolds corresponding to pairwise non-conjugate congruence lattices in a higher rank semisimple group over local fields of zero characteristic is Benjamini--Schramm convergent to the universal cover.

Equations

$${\eta }_n\left(\{g{\Gamma }_n\in G/{\Gamma }_n:g{\Gamma }_n{g}^{-1}\cap Q\subset Z(G)\}\right)\stackrel{n\to \infty }{\to }1$$

$${\mathrm{SL}}_2\left({\mathcal{O}}_F\right)\hookrightarrow \prod _{i=1}^r{\mathrm{SL}}_2\left(\mathbb{R}\right)\times \prod _{i=1}^s{\mathrm{SL}}_2\left(\mathbb{C}\right).$$

$${\nu }_{\Gamma }=\frac{1}{\mathrm{vol}(G/\Gamma )}\sum _{\pi \in \hat{G}}m(\pi ,\Gamma ){\delta }_{\pi }$$

$$\Vert {\pi }_n(\nu {)}_{|L_0^2(X,{\mu }_n)}{\Vert }_n\le \beta$$

$$a_n=\int f\text{d}{\mu }_n,f=f_n^{\prime }+a_n{1}_X$$

$$\begin{gathered} \Vert \pi (\nu )f\Vert =\underset{n}{\lim }\Vert \pi (\nu )f{\Vert }_n\le \underset{n}{\mathrm{limsup}}\Vert \pi (\nu )f_n^{\prime }{\Vert }_n+\underset{n}{\lim }|a_n|\le \\ \le \beta \underset{n}{\mathrm{limsup}}\Vert f_n^{\prime }{\Vert }_n\le \beta \left(\underset{n}{\lim }\Vert f{\Vert }_n+\underset{n}{\lim }|a_n|\right)=\beta \Vert f\Vert . \end{gathered}$$

$$\underset{{\mathfrak{a}}^{\prime }⊲F(S)}{⨁}{L}^2\left(\prod _{v\in S}{\mathbb{H}}_1(F_v)/{\mathbb{H}}_1({\mathfrak{a}}^{\prime })\right)$$

$${\mathbb{H}}_1\cong {\mathrm{SL}}_1\left(D\right)$$

$$H=K_1{T}_1{K}_1\text{and}G=KTK.$$

$${\phi }^g(K_1)\subset K,{\phi }^g(T_1)\subset T\text{and}{\phi }^g({\mathbb{U}}_{\alpha })\subset \mathbb{U}.$$

$$|⟨\beta ,{\phi }^g\circ \lambda ⟩|\le 2$$

$${\phi }^g(K_1)\subset \bigcup _{i=0}^l{K}_i^{\prime },{\phi }^g(T_1)\subset T,\text{and}{\phi }^g({\mathbb{U}}_{\alpha })\subset \mathbb{U}.$$

$$G/\Gamma \to \text{Sub}\left(G\right),g\Gamma \mapsto g\Gamma g^{-1}.$$

Full text

1. Introduction

A semisimple analytic group $G$ is defined as follows. Let $I$ be a finite index set. Assume that $k_{i}$ is a zero characteristic local field and $\mathbb{G}_{i}$ is a connected simply-connected $k_{i}$-isotropic almost $k_{i}$-simple linear $k_{i}$-group for every $i\in I$. Denote $G_{i}=\mathbb{G}_{i}(k_{i})$ so that in particular $G_{i}$ is an almost simple non-compact linear group admitting a $k_{i}$-analytic structure. Let $G=\prod_{i\in I}G_{i}$.

Definition**.**

A sequence of lattices $\left(\Gamma_{n}\right)_{n\in\mathbb{N}}$ in $G$ is called weakly central111Of course, this definition makes sense for any locally compact group. if for every compact subset $Q\subset G$ we have that

[TABLE]

where $\eta_{n}$ is the $G$-invariant probability measure on $G/\Gamma_{n}$ for each $n\in\mathbb{N}$.

This note is dedicated to establishing the following result.

Theorem 1**.**

Assume that $\lvert I\rvert\geq 2$. Then every sequence of pairwise non-conjugate congruence lattices in $G$ is weakly central.

Recall that by the celebrated theorem of Margulis every irreducible lattice in $G$ is arithmetic whenever $\lvert I\rvert\geq 2$. A congruence lattice is a particular kind of an irreducible arithmetic lattice containing a principal congruence subgroup. See §3 for a precise definition of this notion. In particular, whenever lattices in $G$ are known to satisfy the congruence subgroup property a stronger formulation of Theorem 1 is made possible.

We remark that if $\lvert I\rvert=1$ and $\mathrm{rank}(G)\geq 2$ then every sequence of pairwise non-conjugate lattices is weakly central by the main results of [1, 12]. The recent works of Raimbault [21] and Fraczyk [10] establish closely related results for congruence lattices in the rank one groups $\mathrm{SL}_{2}\left(\mathbb{R}\right)$ and $\mathrm{SL}_{2}\left(\mathbb{C}\right)$. We also mention [1, §5] dealing with congruence subgroups in a fixed uniform arithmetic lattice.

Benjamini–Schramm convergence

The semisimple analytic group $G$ is acting by isometries on a contractible non-positively curved metric space $X$, as follows. Let $X_{i}$ be the symmetric space or Bruhat-Tits building associated to $G_{i}$ for every $i\in I$, depending on whether $k_{i}$ is Archimedean or not. Take $X=\prod_{i\in I}X_{i}$ equipped with the product metric.

Let $\left(\Gamma_{n}\right)$ be a sequence of lattices in $G$. The following geometric notion is equivalent to saying that $(\Gamma_{n})$ is weakly central. See [12, §3] for more details.

Definition**.**

The orbifolds $\Gamma_{n}\backslash X$ Benjamini–Schramm converge to $X$ if for every radius $0<R<\infty$ the probability that an $R$-ball in $\Gamma_{n}\backslash X$ with base point taken uniformly at random is contractible tends to one as $n\to\infty$.

As an example, we provide a geometric application of Theorem 1 to arithmetic orbifolds, relying on the congruence subgroup property [23].

Corollary 2**.**

Let $F$ be a number field with ring of integers $\mathcal{O}_{F}$. Assume that $F$ has $r$ real embeddings and $2s$ complex embeddings with $r+s\geq 2$. Consider the irreducible arithmetic lattice

[TABLE]

Let $X=(\mathbb{H}^{2})^{r}\times(\mathbb{H}^{3})^{s}$ be a product of two and three dimensional hyperbolic spaces. Then the orbifolds corresponding to any sequence of distinct finite-index subgroups of $\mathrm{SL}_{2}\left(\mathcal{O}_{F}\right)$ are Benjamini-Schramm convergent to $X$.

The conclusion of Corollary 2 continues to hold if one moreover varies $F$ among all number fields with the same signature.

On properties $(T)$ and $(\tau)$

Benjamini-Schramm convergence for lattices was first investigated in [1]. It is shown in [1] that any sequence of pairwise non-conjugate irreducible lattices in a semisimple Lie group with high rank and property $(T)$ is weakly central. General local fields are dealt with in [12]. These proofs rely on property $(T)$, most crucially in order to invoke the Stuck–Zimmer theorem [16, 24].

Our approach is to make use of property $(\tau)$ instead, avoiding the Stuck–Zimmer theorem which is presently unknown in the absence of property $(T)$. More precisely, we rely on property $(\tau)$ with respect to congruence lattices. This is sometimes called the Selberg property as it generalizes his famous theorem on congruence subgroups of the modular group. It is crucial that Selberg’s property is uniform with respect to all the congruence lattices inside $G$ — see Theorem A.

In addition, we rely on topological properties of the Chabauty space of semisimple analytic groups recently established by Gelander and the author [12], thereby replacing yet another argument of [1] which previously required property $(T)$.

Spectral gap and essentially free actions

Towards proving Theorem 1 we study a Borel $G$-space obtained by taking a certain limit with respect to a sequence of congruence lattices. Selberg’s property implies that this limiting $G$-space has spectral gap — see §2 for a discussion of this notion. The following theorem allows us to deduce that such an action is essentially free, provided that $\lvert I\rvert\geq 2$.

Theorem 3**.**

Let $G$ be a product of at least two locally compact simple groups and $X$ a Borel $G$-space admitting an invariant probability measure $\mu$. Assume that $X$ is properly ergodic, irreducible and has spectral gap. Then $(X,\mu)$ is essentially free.

We do not claim any originality for Theorem 3 — it is a formal corollary of the well-known work of Bader–Shalom [3] and the fact that an action with spectral gap is not weakly amenable [9]. We state it here merely as an observation and in the hope that it may prove useful in other situations as well.

Notice that for Theorem 3 to be applicable in our situation the limiting $G$-space has to be irreducible. Indeed, this follows from a key feature of Theorem A establishing spectral gap with respect to each simple factor of $G$ individually.

An application — convergence of Plancherel measures

Let $\nu^{G}$ denote the Plancherel measure on the unitary dual $\widehat{G}$ of $G$. For every uniform lattice $\Gamma$ in $G$, the quasi-regular representation $\rho_{\Gamma}$ of $G$ in $L^{2}(G/\Gamma)$ decomposes as a direct sum of irreducible representations. Every irreducible representation $\pi\in\widehat{G}$ appears in $\rho_{\Gamma}$ with finite multiplicity $m(\pi,\Gamma)$. The corresponding relative Plancherel measure is

[TABLE]

Combining Theorem 1 with [1, 1.2] and [12, 1.3] we obtain a generalization of one of the main results of [1, 12] regarding convergence of relative Plancherel measures.

Corollary 4**.**

Assume that $\lvert I\rvert\geq 2$. Let $\Gamma_{n}$ be any sequence of pairwise non-conjugate uniformly discrete torsion-free congruence lattices in $G$. Then $\nu_{\Gamma_{n}}(E)\xrightarrow{n\to\infty}\nu^{G}(E)$ for every relatively quasi-compact $\nu^{G}$-regular subset $E\subset\widehat{G}$.

Further applications of Corollary 4 to limit multiplicities formulas and normalized Betti numbers [1, 1.3,1.4] carry over to our setting as well.

Acknowledgements

I would like to thank Tsachik Gelander for his guidance and many useful discussions about the ideas of [1]. I would like to thank Nicolas Bergeron for bringing to my attention the important fact that Clozel’s theorem [7] holds uniformly with respect to all congruence lattices. I would like to thank Uri Bader and Alex Lubotzky for several useful discussions and remarks.

2. Uniform spectral gap

Let $G$ be a compactly generated locally compact group and $(X,\mu)$ a Borel $G$-space with an invariant probability measure. Recall that the $G$-action on $X$ has spectral gap if the Koopman representation of $G$ in the space $L^{2}_{0}(X,\mu)$ of functions with zero integral does not almost admit invariant vectors.

Definition**.**

A sequence $(X_{n},\mu_{n})$ of Borel $G$-spaces with invariant probability measures has uniform spectral gap if the natural representation of $G$ on $\oplus_{n}L^{2}_{0}(X_{n},\mu_{n})$ does not almost admit invariant vectors.

For example, if $G$ has property $(T)$ then any family of ergodic $G$-invariant probability measures has uniform spectral gap. More generally, such uniformity is useful when passing to weak-$*$ limits of probability measures on a given compact $G$-space.

Proposition 5**.**

Let $X$ be a compact $G$-space and $\mu_{n}$ a sequence of invariant Borel probability measures on $X$ with uniform spectral gap. If $\mu$ is a weak-$*$ limit of the sequence $\mu_{n}$ then $\mu$ has spectral gap.

Let $\pi$ and $\pi_{n}$ denote the Koopman representations on the Hilbert spaces $L^{2}(X,\mu)$ and $L^{2}(X,\mu_{n})$ for $n\in\mathbb{N}$. Let $\lVert\cdot\rVert$ and $\lVert\cdot\rVert_{n}$ denote the norms coming from the Hilbert space structure on these spaces.

Proof of Proposition 5.

Let $\nu$ be a probability measure on $G$ which is symmetric, absolutely continuous with respect to the Haar measure and such that $\mathrm{supp}(\nu*\nu)$ is a generating set for $G$. Uniform spectral gap for the measures $\mu_{n}$ means that

[TABLE]

for some constant $0<\beta<1$ and all $n\in\mathbb{N}$. This fact is established in [4, G.4.2].

We claim that $\lVert\pi(\nu)_{|L^{2}_{0}(X,\mu)}\rVert\leq\beta$ as well. In estimating the norm of a continuous operator we may restrict attention to a dense subspace. Consider any non-zero continuous function $f\in C(X)\cap L^{2}_{0}(X,\mu)$. Note that $\pi(\nu)f\in C(X)\cap L^{2}_{0}(X,\mu)$ as well. Denote

[TABLE]

so that $f^{\prime}_{n}\in C(X)\cap L^{2}_{0}(X,\mu_{n})$ for all $n\in\mathbb{N}$.

Since $\mu$ is a weak-$*$ limit of the measures $\mu_{n}$ we have that $\lim_{n}a_{n}=\int f\,\textrm{d}\mu=0$. To estimate the operator norm of $\pi(\nu)$ on the space $L^{2}_{0}(X,\mu)$ we calculate

[TABLE]

Therefore the $G$-space $(X,\mu)$ has spectral gap as well [4, G.4.2]. ∎

Assume that $G$ splits as a direct product $G=G_{1}\times\cdots\times G_{k}$ of $k$-many factors. It is natural to consider the restriction of the $G$-action to each factor $G_{i}$ individually. For instance, the $G$-action is irreducible if each $G_{i}$ is acting ergodically.

Definition**.**

$(X,\mu)$* has strong spectral gap if the restricted action of each factor has spectral gap. A sequence $(X_{n},\mu_{n})$ of Borel probability $G$-spaces has strong uniform spectral gap if these restricted actions have uniform spectral gap.*

Since spectral gap implies ergodicity, strong spectral gap implies irreducibility.

3. Congruence lattices and the Selberg property

Let $G$ be semisimple analytic group. As in the first paragraph of the introduction $G$ is the direct product of the analytic groups $G_{i}=\mathbb{G}_{i}(k_{i})$ where every $\mathbb{G}_{i}$ is a connected almost $k_{i}$-simple222A linear $k$-group $\mathbb{G}$ is almost $k$-simple if $\mathbb{G}$ is semisimple and admits no proper normal $k$-subgroups of positive dimension. linear group and $k_{i}$ a local field for every index $i\in I$.

Congruence lattices

Let $F$ be an algebraic number field and $\mathbb{H}$ an absolutely almost simple333A linear $k$-group $\mathbb{G}$ is absolutely almost simple if $\mathbb{G}$ is almost simple over the algebraic closure of $k$. linear $F$-group. Let $R$ denote the inequivalent Archimedean valuations on the field $F$ such that $\mathbb{H}(F_{v})$ is non-compact for $v\in R$. Assume that there is a finite set of valuations $S$ with $R\subset S$ and a bijection $\iota:I\to S$ so that $k_{i}\cong F_{\iota(a)}$ and $\mathbb{G}_{i}$ is $k_{i}$-isomorphic to $\mathbb{H}$ for all $i\in I$. In particular we may identify $G$ with $\prod_{v\in S}\mathbb{H}(F_{v})$.

Let $F(S)$ denote the ring of $S$-integers in the field $F$. The group $\mathbb{H}(F(S))$ is an irreducible lattice in $G$ by the Borel–Harish-Chandra theorem [20, Theorem 4.8]. Given a non-zero ideal $\mathfrak{a}$ in the ring $F(S)$ let $\mathbb{H}(\mathfrak{a})$ denote the kernel of the natural map $\mathbb{H}(F(S))\to\mathbb{H}(F(S)/\mathfrak{a})$. The subgroup $\mathbb{H}(\mathfrak{a})$ is called a principal congruence subgroup.

Definition**.**

A congruence lattice is any lattice in $G$ containing some $\mathbb{H}(\mathfrak{a})$ as above.

The following is essentially a reformulation of the well-known Selberg’s property.

Theorem A** (Selberg’s property).**

Let $G$ be a semisimple analytic group. Then the family of $G$-spaces $G/\Gamma$ with normalized probability measures and $\Gamma$ ranging over the congruence lattices in $G$ has strong uniform spectral gap.

We begin our discussion of Theorem A with a few preliminary remarks.

• •

The two $G$-representations $L^{2}(X,\mu)$ and $L^{2}(X,\alpha\mu)$ are equivalent for every $\alpha>0$, so that renormalizing a finite measure on a Borel $G$-space has no effect on spectral gap.

• •

Similarly, the two $G$-representations $L^{2}(G/\Gamma)$ and $L^{2}(G/\Gamma^{g})$ are equivalent for every $g\in G$.

• •

For a pair of lattices $\Gamma,\Gamma^{\prime}$ with $\Gamma\leq\Gamma^{\prime}$ the $G$-representation $L^{2}(G/\Gamma^{\prime})$ is contained in $L^{2}(G/\Gamma)$.

In light of these remarks we may restrict our attention to the situation of the lattice $\mathbb{H}(\mathfrak{a})$ inside $\prod_{v\in S}\mathbb{H}(F_{v})$ while making sure that the resulting strong spectral gap is independent of the field $F$, the group $\mathbb{H}$ and the ideal $\mathfrak{a}$.

On Selberg’s property

The existence of spectral gap for congruence subgroups of $\mathrm{SL}_{2}\left(\mathbb{Z}\right)$ regarded as lattices in $\mathrm{SL}_{2}\left(\mathbb{R}\right)$ is essentially Selberg’s classical $\lambda_{1}\geq\frac{3}{16}$ theorem [22]. The Archimedean case444Note that this case already suffices for the purpose of our Corollary 2. where $\mathbb{H}$ is still $\mathrm{SL}_{2}$, $F$ is any number field and $S$ consists of infinite places is treated in [25]. The remaining case of $\mathbb{H}=\mathrm{SL}_{2}$ and $S$ an arbitrary set of places follows from the work of Gelbart–Jacquet [13].

The Burger–Sarnak method [6] allows to go beyond $\mathrm{SL}_{2}$. This method was extended by Clozel–Ullmo [8], in particular covering the $p$-adic case. We refer the reader to the useful discussion on [18, §4.2].

Theorem B** (Burger–Sarnak, Clozel–Ullmo).**

Let $\mathbb{H}_{1}$ be a semi-simple $F$-subgroup of $\mathbb{H}$. Then for every valuation $v\in S$ the restriction of $L^{2}\left(G/\mathbb{H}(\mathfrak{a})\right)$ to $\mathbb{H}_{1}(F_{v})$ is weakly contained in

[TABLE]

where the direct sum is taken over all non-zero ideals in $F(S)$.

Clozel made the final contribution towards Selberg’s property by dealing with arbitrary absolutely simple groups. In fact Theorem A is essentially equivalent to [7, Thm. 3.1]. Clozel’s proof for a general $F$-group $\mathbb{H}$ depends on whether it is $F$-isotropic or not. If $\mathrm{rank}_{F}(\mathbb{H})\geq 1$ then $\mathbb{H}$ is known to admit $\mathrm{SL}_{2}$ as a $F$-subgroup [19, I.1.6.3]. In that case one may rely on [13] and Theorem B.

The main effort of [7] is in dealing with the anisotropic case, as follows. If $\mathbb{H}$ is $F$-anisotropic then it admits a $F$-subgroup $\mathbb{H}_{1}$ with $\mathrm{rank}_{F_{v}}(\mathbb{H}_{1})=1$ and such that

[TABLE]

where $D$ is a division algebra of degree $p^{2}$ over $F$ or over a quadratic extension of $F$ in the first and second cases, respectively, and $p$ is a prime [7, 1.1]. The careful analysis of [7, §3.2] reveals that the case of $\mathrm{SL}_{1}\left(D\right)$ can be reduced to $\mathrm{SL}_{2}$. Similarly, the case of $\mathrm{SU}(D,*)$ can be reduced either to $\mathrm{SL}_{2}$, to $\mathrm{SU}(3,F_{v})$ with $v$ finite or to $\mathrm{SU}(n,1)$ with $v$ infinite. The parameter $n$ is clearly bounded since $\mathbb{H}_{1}(F_{v})$ embeds in $G$. Clozel then establishes spectral gap directly for congruence lattices in these last two families of rank one groups, and concludes relying on Theorem B.

Uniformity of spectral gap

Note that while [7, Theorem 3.1] is stated with respect to a fixed algebraic number field $F$ and group $\mathbb{H}$, the resulting spectral gap for the subgroup $\mathbb{H}_{1}(F_{v})$ turns out to be independent of any such choices.

To see this, observe that there are only finitely many possibilities for the group $\mathbb{H}_{1}(F_{v})$ and that the validity of Theorem A for these implies the same for $G$. The fact that $\mathbb{H}_{1}(F_{v})$ regarded as a subgroup of $G$ depends on the chosen $F$-structure needs to be taken into account, relying on the above preliminary remarks and the following lemma.

Lemma 6**.**

Let $\mathbb{H}$ and $\mathbb{G}$ be a pair of almost $k$-simple linear groups over a local field $k$ of zero characteristic. Assume that $\mathrm{rank}_{k}(\mathbb{H})=1$ and denote $H=\mathbb{H}(k)$ and $G=\mathbb{G}(k)$. Then for every compact subset $Q_{1}\subset H$ there is a compact $Q\subset G$ such that every $k$-homomorphism $\varphi:\mathbb{H}\to\mathbb{G}$ satisfies $\varphi(Q_{1})^{g}\subset Q$ for some $g\in G$.

Proof.

Consider the Archimedean case first. Weil’s restriction of scalars allows us to assume without loss of generality that the local field $k$ is $\mathbb{R}$.

Let $\mathbb{T}_{1}$ and $\mathbb{T}$ be maximal $k$-split tori in the groups $\mathbb{H}$ and $\mathbb{G}$ respectively. Let $\mathbb{B}$ be a minimal parabolic $k$-subgroup of $\mathbb{G}$ containing the torus $\mathbb{T}$ and with unipotent $k$-subgroup $\mathbb{U}$. Since $\mathrm{rank}_{k}(\mathbb{T}_{1})=1$ the $k$-root system of $\mathbb{H}$ relative to $\mathbb{T}_{1}$ has type $\mathrm{A}_{1}$ and $\Phi_{k}(\mathbb{H},\mathbb{T}_{1})=\{\alpha,-\alpha\}$ for some $k$-character $\alpha\in X(\mathbb{T}_{1})$. Let $\mathbb{U}_{\alpha}$ be the unipotent $k$-subgroup of $\mathbb{H}$ corresponding to the root $\alpha$. This means that the Lie subalgebra $\mathrm{Lie}(\mathbb{U}_{\alpha})$ is the weight space of $\alpha$ in the adjoint representation on the Lie algebra $\mathrm{Lie}(\mathbb{H})$.

Denote $T_{1}=\mathbb{T}_{1}(k)$ and $T=\mathbb{T}(k)$. Let $K_{1}$ and $K$ be maximal compact subgroups of $H$ and of $G$ respectively, admitting corresponding Cartan decompositions

[TABLE]

Let $\varphi:\mathbb{H}\to\mathbb{G}$ be any $k$-homomorphism. We claim that there is an element $g\in G$ such that

[TABLE]

To establish the claim we argue as follows. All maximal compact subgroups of $G$ are conjugate [20, 3.10]. Therefore there is an element $g_{1}\in G$ such that $\varphi^{g_{1}}(K_{1})\subset K$. Let $T^{\prime}$ be any maximal $k$-split torus of $G$ containing $\varphi^{g_{1}}(T_{1})$. There is an element $g_{2}\in K$ satisfying $(T^{\prime})^{g_{2}}=T$, see [5, 20.9, 24.7] or [15, 6.51]. The relative Weyl group of the torus $\mathbb{T}$ is transitive on the set of minimal parabolic $k$-subgroups containing $\mathbb{T}$ [5, 21.3], and every element of this Weyl group can be realized in the maximal compact subgroup $K$ [15, 6.57]. Hence there is an element $g_{3}\in K\cap\mathrm{N}_{G}(T)$ such that $\varphi^{g}(\mathbb{U}_{\alpha})\subset\mathbb{U}$ where $g=g_{3}g_{2}g_{1}$. This concludes the claim.

Let $\lambda\in X_{*}(\mathbb{T}_{1})$ be the one-parameter $k$-subgroup of $\mathbb{T}_{1}$ satisfying $\left<\alpha,\lambda\right>=2$. The $k$-split torus $\mathbb{T}$ acts on the unipotent $k$-subgroup $\mathbb{U}$ by conjugation. In fact $\mathbb{U}$ is $\mathbb{T}$-equivariantly $k$-isomorphic as a $k$-variety to its Lie algebra $\mathrm{Lie}(\mathbb{U})$ [5, 15.13]. This implies that

[TABLE]

for all $k$-roots $\beta\in\Phi_{k}(\mathbb{G},\mathbb{T})$. Here $g\in G$ is an element as in the above claim.

The compactness of $Q_{1}$ implies that $Q_{1}\subset K_{1}Q^{\prime}_{1}K_{1}$ for some compact subset $Q^{\prime}_{1}\subset T_{1}$. By the upper bound on $|\left<\beta,\varphi^{g}\circ\lambda\right>|$ for all $\beta\in\Phi_{k}(\mathbb{G},\mathbb{T})$ there is a compact subset $Q^{\prime}\subset T$ depending on $Q^{\prime}_{1}$ such that $\varphi^{g}(Q^{\prime}_{1})\subset Q^{\prime}$. The Archimedean case of the proof follows letting $Q=KQ^{\prime}K$.

We now deal with the non-Archimedean case. Weil’s restriction of scalars allows us to assume without loss of generality that the local field $k$ is $\mathbb{Q}_{p}$ for some prime number $p$.

Let the notations $\mathbb{T}_{1},T_{1},\mathbb{U}_{\alpha},\mathbb{T},T,\mathbb{B}$ and $\mathbb{U}$ be exactly as in the Archimedean case. Let $K_{1}$ be a maximal compact subgroup of $H$ admitting a Cartan decomposition $H=K_{1}T_{1}K_{1}$. Let $B$ be an Iwahori subgroup [14] of $G$ corresponding to the minimal parabolic subgroup $\mathbb{B}$. Let $K^{\prime}_{0},\ldots,K^{\prime}_{l}$ be the maximal compact subgroups of $G$ containing $B$. In particular $l=\mathrm{rank}_{k}(\mathbb{G})$. These are precisely the representatives of the conjugacy classes of all maximal compact subgroups in $G$. There are corresponding Cartan decompositions $G=K^{\prime}_{i}TK^{\prime}_{i}$ for all $i\in\{0,\ldots,l\}$. For details see e.g. [20, 3.13, 3.14].

Let $\varphi:\mathbb{H}\to\mathbb{G}$ be any $k$-homomorphism. In analogy with the Archimedean case, we claim that there is an element $g\in G$ such that

[TABLE]

To see this we argue as follows. There is an element $g_{1}\in G$ such that $\varphi^{g_{1}}(K_{1})\subset K^{\prime}_{i}$ for some $i\in\{0,\ldots,l\}$. Let $T^{\prime}$ be any maximal $k$-split torus of $G$ containing $\varphi^{g_{1}}(T_{1})$. The Bruhat–Tits building associated to $G$ is strongly transitive [2, 6.56]. Therefore there is some $g_{2}\in B$ such that $(T^{\prime})^{g_{2}}=T$. In particular $(K^{\prime}_{i})^{g_{2}}=K^{\prime}_{j}$ for some $j\in\{0,\ldots,l\}$. Every element of the restricted Weyl group of the torus $\mathbb{T}$ is realized in the maximal compact subgroup $K^{\prime}_{j}$ [20, p. 150]. There is an element $g_{3}\in K^{\prime}_{j}\cap\mathrm{N}_{G}(T)$ satisfying $\varphi^{g}(\mathbb{U}_{\alpha})\subset\mathbb{U}$ with $g=g_{3}g_{2}g_{1}$. The claim follows.

The final part of the proof is exactly as in the Archimedean case. We may take $K^{\prime}=\bigcup_{i=0}^{l}K^{\prime}_{i}$ and $Q=K^{\prime}Q^{\prime}K^{\prime}$ for some sufficiently large compact subset $Q^{\prime}\subset T$ depending on $Q_{1}$. ∎

4. Weakly amenable actions and Theorem 3

Let $G$ be a locally compact group and $(X,\mu)$ a Borel $G$-space with an invariant probability measure. We use spectral gap to deduce essential freeness for such an action and establish Theorem 3.

Recall that a $G$-space $(X,\mu)$ is weakly amenable if the orbital equivalence relation generated by the action is amenable; see e.g. [26, Section 4.3] and [24] for details.

Theorem C** (Stuck–Zimmer, Bader–Shalom).**

Let $G$ be a direct product of at least two simple groups. Assume that the $G$-space $(X,\mu)$ is properly ergodic, irreducible and not weakly amenable. Then it is $\mu$-essentially free.

Proof.

See [24] for the classical case of semisimple Lie groups, [16] for semisimple linear groups over local fields and [3] for general locally compact groups. ∎

We conclude that Theorem 3 follows at once by combining Theorem C with the following observation, due to Creutz [9].

Proposition D**.**

Let $G$ be a second countable locally compact group. If the $G$-space $(X,\mu)$ is properly ergodic and weakly amenable then it has no spectral gap.

Roughly speaking, the proof of Proposition D is as follows. A weakly amenable action is orbit equivalent to an action of either $\mathbb{Z}$ or $\mathbb{R}$, depending on whether $G$ is countable or not. The action of these amenable groups admits more than a single invariant mean on $L^{\infty}(X,\mu)$. This fact is invariant under orbit equivalence, so that the action of $G$ admits more than a single invariant mean as well. Therefore the action of $G$ is not strongly ergodic and in particular there can be no spectral gap. See [9, 7.3.1] for complete details and references concerning this argument.

It is interesting to note the similarity of the above theme with the role played by Selberg’s theorem in the solution of the Banach-Ruziewicz problem [17].

5. Invariant random subgroups and a proof of Theorem 1

We now prove Theorem 1 relying on strong uniform spectral gap for congruence lattices. The proof strategy is inspired by [1] and in particular invariant random subgroups555See Gelander’s lecture notes [11] on the Chabauty topology, invariant random subgroups and Benjamini-Schramm spaces. play a key role.

Associated to any second countable locally compact group $G$ is the Chabauty space of closed subgroups, denoted $\textrm{Sub}\left(G\right)$. This is a compact $G$-space with the Chabauty topology and the conjugation action. An invariant random subgroup of $G$ is a $G$-invariant probability measure on $\textrm{Sub}\left(G\right)$. Let $\mathrm{IRS}\left(G\right)$ denote the compact convex space of all invariant random subgroups of $G$ with the weak-$*$ topology.

Associated to any normal subgroup $N\vartriangleleft G$ is the point mass $\delta_{N}\in\mathrm{IRS}\left(G\right)$. More interestingly, to any lattice $\Gamma$ in $G$ we associate $\mu_{\Gamma}\in\mathrm{IRS}\left(G\right)$ obtained by pushing forward the $G$-invariant probability from $G/\Gamma$ to $\textrm{Sub}\left(G\right)$ via the $G$-equivariant mapping

[TABLE]

Let $\Gamma_{n}$ be a sequence of pairwise non-conjugate lattices in $G$ and $\mu$ any weak-$*$ accumulation point of $\mu_{\Gamma_{n}}$ in the space $\mathrm{IRS}\left(G\right)$.

Proposition 7**.**

If $\mu$ is irreducible and essentially transitive then $\mu=\delta_{M}$ for some central subgroup $M$.

Proof.

Since $(\textrm{Sub}\left(G\right),\mu)$ is essentially transitive there is a closed subgroup $H\leq G$ such that $\mu$ is supported on the conjugacy class of $H$ and the $G$-space $(\textrm{Sub}\left(G\right),\mu)$ is isomorphic to the homogeneous $G$-space $G/N_{G}(H)$. Therefore the subgroup $N_{G}(H)$ is of finite covolume in $G$.

The factors $G_{i}$ of $G$ are non-compact and the fields $k_{i}$ have zero characteristic so that the density theorem of Borel applies. As in [19, II.6.2.(b)] it can used to deduce that $N_{G}(H)=M\times\Gamma$ where $M\vartriangleleft G$ is a normal subgroup with $G=M\times M^{\prime}$ and $\Gamma$ is a lattice in $M^{\prime}$. Since the normal subgroup $M$ acts trivially on $G/M$ the irreducibility of the action implies either that $N_{G}(H)$ is an irreducible lattice in $G$ or that $N_{G}(H)=G$. In the second case $\mu$ is equal to $\delta_{M}$.

We deal with these two possibilities separately, relying on results from [12]. If $N_{G}(H)$ is a lattice in $G$ then $H$ must be an irreducible lattice as well by the normal subgroup theorem of Margulis [19, IV]. Every irreducible lattice in $G$ admits a Chabauty open neighborhood in $\textrm{Sub}\left(G\right)$ consisting of conjugates [12, 1.9], so that the corresponding point $\mu_{H}$ is isolated in the space of extreme points of $\mathrm{IRS}\left(G\right)$.

A non-discrete normal subgroup of $G$ does not belong to the closure of the Chabauty subspace of discrete subgroups [12, 6.7]. Therefore the second case where $\mu=\delta_{M}$ is impossible unless $M$ is central, as required. ∎

Assume moreover that the $\Gamma_{n}$’s are congruence lattices. We claim that $\mu$ is indeed equal to $\delta_{M}$ for some central subgroup $M$. Theorem A implies that the sequence $\mu_{\Gamma_{n}}$ of $G$-invariant Borel probability measures on the compact $G$-space $\textrm{Sub}\left(G\right)$ has strong uniform spectral gap. By Proposition 5 the $G$-space $(\textrm{Sub}\left(G\right),\mu)$ has strong spectral gap as well. In particular $\mu$ is both irreducible and has spectral gap.

By the argument on [24, p. 729] we may assume to begin with that $G$ has trivial center. The $G$-action on $(\textrm{Sub}\left(G\right),\mu)$ is certainly not essentially free. The contrapositive to Theorem 3 implies that it is not properly ergodic. Therefore the $G$ action on $(\textrm{Sub}\left(G\right),\mu)$ must be essentially transitive and Proposition 7 applies.

The following three modes of convergence are all equivalent for semisimple analytic groups over local fields of zero characteristic [12] — weak-$*$ convergence of $\mu_{\Gamma_{n}}$ to a central subgroup in $\mathrm{IRS}\left(G\right)$, Benjamini–Schramm convergence of $\Gamma_{n}\backslash X$ to $X$ and the fact that the sequence $\Gamma_{n}$ is weakly central.

The proof of Theorem 1 is now complete.