Periodic points of algebraic actions of discrete groups

TL;DR

This paper proves that Noetherian algebraic actions of finitely generated virtually nilpotent groups on compact abelian groups have a dense set of periodic points, advancing understanding of dynamical properties of such group actions.

Contribution

It establishes the density of periodic points for Noetherian algebraic actions of finitely generated virtually nilpotent groups, a new result in the study of algebraic group actions.

Findings

Noetherian actions have dense periodic points

Finitely generated virtually nilpotent groups exhibit this property

Advances understanding of algebraic dynamical systems

Abstract

Let $\Gamma$ be a countable group. A $\Gamma$-action on a compact abelian group $X$ by continuous automorphisms of $X$ is called Noetherian if the dual of $X$ is Noetherian as a ${\mathbb Z}(\Gamma)$-module. We prove that any Noetherian action of a finitely generated virtually nilpotent group has a dense set of periodic points.

Equations

$$X=X_0\supset X_1\supset X_2\supset \cdots$$

$$c(\gamma {\gamma }^{{}^{\prime }})=c(\gamma )+\alpha (\gamma )(c({\gamma }^{{}^{\prime }}))$$

$$c_y(\gamma )=\beta (\gamma )(y)-y=\beta (\gamma )(x)-x$$

$$\beta (\gamma ){\beta }^{{}^{\prime }}(\gamma )(1,0)=\beta (\gamma )(1,0)+\beta (\gamma )(0,-\alpha (\gamma {)}^{-1}c(\gamma ))=(1,0).$$

$$\beta ({\gamma }_1{\gamma }_2)(1,0)=(1,c({\gamma }_1{\gamma }_2))=\beta ({\gamma }_1)\beta ({\gamma }_2)(1,0).$$

$${\Lambda }_0=\bigcap _{\gamma \in \Gamma }\gamma \Lambda {\gamma }^{-1}.$$

$$H=\sum _{\gamma \in \Gamma }\alpha (\gamma )(Y_0).$$

$$lk{c}_1(\gamma )=lk(\beta (\gamma )(y-q)-(y-q))=0.$$

$$(f_1,n_1)\cdot (f_2,n_2)=(f_1+n_1\cdot f_2,n_1+n_2).$$

$$\sum _{i=-\infty }^{\infty }f(ki+j)=0(\mathrm{mod}k)\forall j=0,1,\dots ,k-1.$$

$$S^{n_1}(\pi (f_2))=S^{n_1}(c(f_2,n_2))=\alpha (f_1,n_1)(c(f_2,n_2)),$$

$$\Gamma ={\Gamma }_n\supset {\Gamma }_{n-1}\supset \cdots \supset {\Gamma }_0=\{0\}$$

$$h(\Gamma /N)=h(\Gamma )-h(N)<h(\Gamma ).$$

$$c(\gamma )=c^{{}^{\prime }}(\pi (\gamma ))={\alpha }^{{}^{\prime }}(\pi (\gamma ))(x)-x=\alpha (\gamma )(x)-x.$$

$$P\circ c(\gamma )=\beta (\gamma )(p)-p\forall \gamma \in {\Lambda }_1.$$

$$c_2({\gamma }_2\gamma )=c_2(\gamma {\gamma }_2)=c_2(\gamma )+\alpha (\gamma )(c_2({\gamma }_2))=c_2(\gamma ).$$

Full text

1. Introduction

Let $\Gamma$ be a countable group. An algebraic $\Gamma$-action is an action $\alpha$ of $\Gamma$ on a compact metrizable abelian group $X$ by continuous automorphisms of $X$. By duality theory, any such action induces a $\Gamma$-action on ${\widehat{X}}$, the dual of $X$, by automorphisms of ${\widehat{X}}$. Hence ${\widehat{X}}$ can be viewed as a $\mathbb{Z}[\Gamma]$-module, where $\mathbb{Z}[\Gamma]$ is the integral group ring of $\Gamma$. The action $\alpha$ is called Noetherian if ${\widehat{X}}$ is Noetherian as a $\mathbb{Z}[\Gamma]$-module. Equivalently, $\alpha$ is Noetherian if any decreasing sequence

[TABLE]

of closed $\Gamma$-invariant subgroups stabilizes.

For $\Gamma=\mathbb{Z}^{d}$, the study of such actions was initiated in [5], and since then these systems have been extensively studied (see [9] for a comprehensive account). For general $\Gamma$, several important results about entropy and homoclinic points of Noetherian $\Gamma$-actions have been proved in recent years ([1], [2], [3], [6]). However, unlike the commutative case, many basic dynamical properties are only partly understood in the general situation.

In this paper, we study periodic points of Noetherian $\Gamma$-actions. Recall that a point $x\in X$ is periodic if the $\Gamma$-orbit of $x$ is finite. It is known that when $\Gamma=\mathbb{Z}^{d}$, for any Noetherian $\Gamma$-action $(X,\alpha)$ the space $X$ contains a dense set of periodic points ([9], Theorem 5.7). However, the proof uses tools from commutative algebra and does not generalize to actions of non-abelian groups. The question whether Noetherian actions of all residually finite groups admit a dense set of periodic points was raised in [7, Problem 8.5].

In this paper we show that for algebraic actions of countable groups, the density of periodic points is related to vanishing of certain cohomology groups. As a consequence, we obtain the following :

Theorem 1.1**.**

Let $\Gamma$ be a finitely generated virtually nilpotent group and let $\alpha$ be a Noetherian action of $\Gamma$ on a compact metrizable abelian group $X$. Then $X$ contains a dense set of $\alpha$-periodic points.

As another application we show that there exist Noetherian actions of finitely generated residually finite groups that do not have a dense set of periodic points, thus giving a negative answer to the above question in the most general case.

2. Virtual first cohomology groups

Let $\alpha$ be an action of a countable group $\Gamma$ on a compact metrizable group $X$ by continuous automorphisms of $X$. A *virtual 1-cocycle * of $\alpha$ is a map $c$ from a finite index subgroup $\Lambda\subset\Gamma$ to $X$ that satisfies the equation

[TABLE]

for all $\gamma,\gamma^{{}^{\prime}}\in\Lambda$. Two virtual 1-cocycles $c_{1}:\Lambda_{1}\rightarrow X$ and $c_{2}:\Lambda_{2}\rightarrow X$ are equivalent if there exists a finite index subgroup $\Lambda\subset\Lambda_{1}\cap\Lambda_{2}$ such that $c_{1}(\gamma)=c_{2}(\gamma)$ for all $\gamma\in\Lambda$. For any $x\in X$, the map $c_{x}:\Gamma\rightarrow X$ defined by $c_{x}(\gamma)=\alpha(\gamma)(x)-x$ is a 1-cocycle. A virtual 1-cocycle $c:\Lambda\rightarrow X$ is said to be a *virtual coboundary * if it is equivalent to $c_{x}$ for some $x\in X$. We will call two virtual 1-cocycles $c_{1}:\Lambda_{1}\rightarrow X$ and $c_{2}:\Lambda_{2}\rightarrow X$ cohomologous if there exists a finite index subgroup $\Lambda\subset\Lambda_{1}\cap\Lambda_{2}$, and a virtual coboundary $c:\Lambda\rightarrow X$ such that $c_{1}(\gamma)-c_{2}(\gamma)=c(\gamma)$ for all $\gamma\in\Lambda$. It is easy to see that the equivalence classes of virtual 1-cocyles is a group with respect to the pointwise addition, and the equivalence classes of virtual coboundaries form a subgroup. We will denote the quotient group by $H_{v}^{1}(\alpha)$ and call it the *virtual first cohomology group * of $\alpha$.

Let $(Y,\beta)$ be an algebraic $\Gamma$-action, and let $X\subset Y$ be a closed $\beta$-invariant subgroup such that $Y/X$ is finite. Let $\alpha$ denote the restriction of $\beta$ to $X$ and let $\Lambda\subset\Gamma$ be a finite index subgroup that acts trivially on $Y/X$. For any $y\in Y$ we define a map $c_{y}:\Lambda\rightarrow X$ by $c_{y}(\gamma)=\beta(\gamma)(y)-y$. It is easy to see that $c_{y}$ is a virtual 1-cocycle of $\alpha$.

Proposition 2.1**.**

Let $(Y,\beta)$, $(X,\alpha)$ and $y\in Y$ be as above. Then the virtual 1-cocycle $c_{y}$ is a virtual coboundary if and only if the coset $y+X$ contains a $\beta$-periodic point.

Proof. Suppose $y_{1}\in y+X$ is a $\beta$-periodic point, and $\Lambda_{1}\subset\Lambda$ is a finite index subgroup such that $\beta(\gamma)(y_{1})=y_{1}$ for all $\gamma\in\Lambda_{1}$. Then $x=y-y_{1}\in X$ and

[TABLE]

for all $\gamma\in\Lambda_{1}$. Hence $c_{y}$ is a virtual coboundary of $\alpha$. Conversely, if $c_{y}$ is a virtual coboundary of $\alpha$, then there exists a finite index subgroup $\Lambda_{1}$ and $x\in X$ such that $c_{y}(\gamma)=\beta(\gamma)(x)-x$ for all $\gamma\in\Lambda_{1}$. If $p=y-x$ then $p\in y+X$ and it is fixed by $\beta(\Lambda_{1})$. Hence it is a periodic point. $\Box$

A compact abelian group $X$ is *zero-dimensional * if the connected component containing $0_{X}$ is trivial. From the duality theory it follows that $X$ is zero-dimensional if and only if every element of ${\widehat{X}}$ has finite order.

Proposition 2.2**.**

Let $\Gamma$ be a countable group, and let $\alpha$ be a Noetherian action of $\Gamma$ on a zero-dimensional compact abelian group $X$. Then there exists $k\geq 1$ such that $kx=0$ for all $x\in X$.

Proof. Let $\{\chi_{1},\ldots,\chi_{n}\}\subset{\widehat{X}}$ be a finite set that generates ${\widehat{X}}$ as a ${\mathbb{Z}}(\Gamma)$-module. As every element of ${\widehat{X}}$ has finite order, we can find $k\geq 1$ such that $k\chi_{i}=0$ for $i=1,\ldots,n$. Since for any $m$ the set $N_{m}=\{\chi:m\chi=0\}$ is a $\mathbb{Z}(\Gamma)$-submodule of ${\widehat{X}}$, we deduce that $N_{k}={\widehat{X}}$, i.e., $k\chi=0$ for all $\chi\in{\widehat{X}}$. If $x\in X$ then for all $\chi\in{\widehat{X}}$, $\chi(kx)=k\chi(x)=0$. Hence $kx=0$. $\Box$

Our next proposition shows that all virtual 1-cocycles of Noetherian $\Gamma$-actions on zero-dimensional groups arise in the manner described in Proposition 2.1.

Proposition 2.3**.**

Let $\Gamma$ be a countable group and let $(X,\alpha)$ be a Noetherian action of $\Gamma$ on a zero-dimensional group $X$. Then for any virtual 1-cocycle $c:\Lambda\rightarrow X$ there exists $k\geq 1$ and an algebraic $\Lambda$-action $\beta$ on $Y=({\mathbb{Z}}/k{\mathbb{Z}})\times X$ such that

1. (1)

For all $\gamma\in\Lambda$, $\beta(\gamma)$ fixes the first co-ordinate. 2. (2)

$\beta(\gamma)(0,x)=(0,\alpha(\gamma)(x))$* for all $\gamma\in\Lambda$ and $x\in X$.* 3. (3)

$c_{(1,0)}=c.$**

Proof. Since $\alpha$ is Noetherian and $X$ is zero-dimensional, by the previous proposition there exists $k\geq 1$ such that $kx=0$ for all $x\in X$. For any $z\in Y$ and a continuous automorphism $\tau$ of $X$ there exists a unique homomorphism $\theta:Y\rightarrow Y$ such that $\theta(1,0)=z$ and $\theta(0,x)=(0,\tau(x))$ for all $x\in X$. It is easy to see that $\theta$ is continuous. For $\gamma\in\Lambda$ let $\beta(\gamma)$ denote the unique continuous endomorphism of $Y$ satisfying the following two conditions :

1. (1)

$\beta(\gamma)(0,x)=(0,\alpha(\gamma)(x))$ for all $x\in X$. 2. (2)

$\beta(\gamma)(1,0)=(1,c(\gamma))$.

Similarly we define $\beta^{{}^{\prime}}(\gamma)$ to be the unique continuous endomorphism of $Y$ with the property that $\beta^{{}^{\prime}}(\gamma)(0,x)=(0,\alpha(\gamma)^{-1}(x))$ and $\beta^{{}^{\prime}}(\gamma)(1,0)=(1,-\alpha(\gamma)^{-1}c(\gamma))$. Then

[TABLE]

As $\beta(\gamma)\beta^{{}^{\prime}}(\gamma)$ fixes both $\{0\}\times X$ and the point $(1,0)$, we conclude that $\beta(\gamma)\beta^{{}^{\prime}}(\gamma)=I$. A similar computation shows that $\beta^{{}^{\prime}}(\gamma)\beta(\gamma)=I$. Hence $\beta(\gamma)$ is a continuous automorphism of $Y$ for each $\gamma\in\Lambda$. As $c(\gamma_{1}\gamma_{2})=c(\gamma_{1})+\alpha(\gamma_{1})c(\gamma_{2})$, it follows that

[TABLE]

Since $\beta(\gamma_{1}\gamma_{2})(0,x)=\beta(\gamma_{1})\beta(\gamma_{2})(0,x)$ for all $x\in X$, this imlies that $\beta(\gamma_{1}\gamma_{2})=\beta(\gamma_{1})\beta(\gamma_{2})$. Hence $\beta$ defines an algebraic $\Lambda$-action on $Y$. It is easy to see that $\beta$ satisfies all three conditions stated in the proposition. $\Box$

Lemma 2.4**.**

Let $\Gamma$ be a countable group and let $(X,\alpha)$ be a Noetherian $\Gamma$-action such that $X$ admits non-zero periodic points. Then $X$ also contains non-trivial $\alpha$-invariant finite subgroups.

Proof. Let $x\in X$ be a non-zero $\alpha$-periodic point and let $\Lambda\subset\Gamma$ be the stabilizer of $x$. Since the orbit of $x$ is finite, $\Lambda$ is a finite index subgroup of $\Gamma$. We define a subgroup $\Lambda_{0}\subset\Gamma$ by

[TABLE]

It is easy to see that $\Lambda_{0}\subset\Gamma$ is normal and has finite index. Let $Y$ denote the set of points of $X$ that are fixed by all elements of $\Lambda_{0}$. Then $Y$ is a non-trivial $\alpha$-invariant closed subgroup of $X$. Since $\Lambda_{0}\subset\Gamma$ has finite index, the action $\alpha|_{\Lambda_{0}}$ is also Noetherian. As $\Lambda_{0}$ acts trivially on $Y$, it follows that any collection of closed subgroups of $Y$ has a minimal element. Let $Y_{0}$ be a minimal element of the collection of all non-trivial subgroups of $Y$. Since closed subgroups of $Y_{0}$ are in one to one correspondence with subgroups of ${\widehat{Y}_{0}}$, we deduce that ${\widehat{Y}_{0}}$ does not admit non-trivial subgroups. Therefore ${\widehat{Y}_{0}}$, and hence $Y_{0}$, are isomorphic with ${\mathbb{Z}}/p{\mathbb{Z}}$ for some prime $p$. In particular, $Y_{0}$ is finite. We define $H\subset X$ by

[TABLE]

As $\alpha(\gamma)(Y_{0})=Y_{0}$ for all $\gamma\in\Lambda_{0}$, and $\Lambda_{0}\subset\Gamma$ has finite index, the above sum is finite and $H$ is a well defined closed subgroup of $X$. It is easy to see that $H$ is finite and $\alpha$-invariant. $\Box$

The above lemma is not true if $\alpha$ is not assumed to be Noetherian. For example, suppose $\Gamma$ is an arbitrary countable group and $\alpha$ is the trivial action of $\Gamma$ on $X={\widehat{\mathbb{Q}}}$. Then every point of $X$ is $\alpha$-periodic, but since the group ${\mathbb{Q}}$ does not have non-trivial finite quotients, $X$ does not admit non-trivial finite subgroups.

Theorem 2.5**.**

Let $\Gamma$ be a countable group satisfying :

1. (1)

Every Noetherian $\Gamma$-action $(X,\alpha)$ with $X\neq\{0\}$, has a non-zero $\alpha$-periodic point. 2. (2)

The group $H_{v}^{1}(\alpha)$ is trivial for every Noetherian $\Gamma$-action $(X,\alpha)$ on a zero-dimensional group $X$.

Then every Noetherian action of $\Gamma$ on a compact abelian group (not necessarily zero-dimensional) admits a dense set of periodic points. Conversely, if every Noetherian $\Gamma$-action admits a dense set of periodic points, then both the conditions are satisfied.

Proof. Suppose every Noetherian action of $\Gamma$ admits a dense set of periodic orbits. Then the first condition is automatically satisfied. Let $\alpha$ be a Noetherian $\Gamma$-action on a zero-dimensional group $X$, and let $c:\Lambda\rightarrow X$ be a virtual 1-cocycle of $\alpha$. By Proposition 2.3, there exists $k\geq 1$ and an algebraic $\Lambda$-action $\beta$ on $Y=({\mathbb{Z}}/k{\mathbb{Z}})\times X$ such that $c=c_{(1,0)}$ and $\beta(\gamma)|_{X}=\alpha(\gamma)$ for all $\gamma\in\Lambda$. Since the coset containing $(1,0)$ is an open subset of $Y$, it contains a $\beta$-periodic orbit. From Proposition 2.1 we deduce that $c$ is a virtual coboundary. Hence $H^{1}_{v}(\alpha)=\{0\}$.

Now suppose $\Gamma$ satisfies both the conditions stated above. Let $(Y,\beta)$ be a Noetherian action of $\Gamma$. Let $X\subset Y$ denote the closure of the set of all $\beta$-periodic points. Since the sum of two periodic points is again a periodic point it follows that $X$ is a closed $\beta$-invariant subgroup. Suppose $X\neq Y$. Let ${\bar{\beta}}$ denote the induced action of $\Gamma$ on the quotient $Y/X$. Since $\Gamma$ satisfies the first condition, by Lemma 2.4 there exists a finite ${\bar{\beta}}$-invariant non-zero subgroup $F\subset Y/X$. We define $Y^{{}^{\prime}}=\pi^{-1}(F)$, where $\pi$ is the projection map from $Y$ to $Y/X$. We choose a finite index subgroup $\Lambda\subset\Gamma$ that acts trivially on $F$ under ${\bar{\beta}}$. Let $y\notin X$ be a point in $Y^{{}^{\prime}}$, and let $c_{y}:\Lambda\rightarrow X$ denote the virtual 1-cocycle of $(X,\beta)$ defined by $c_{y}(\gamma)=\beta(\gamma)(y)-y$.

Let $X_{0}$ denote the connected component of $X$. It is easy to see that $X_{0}$ is a $\beta$-invariant closed subgroup of $X$. Let $\beta_{1}$ denote the induced action of $\Lambda$ on $X/X_{0}$. As $X/X_{0}$ is zero-dimensional and $\beta_{1}$ is Noetherian, applying Proposition 2.2 we deduce that there exists $l\geq 2$ such that $lp\in X_{0}$ for all $p\in X$. Let $k\geq 2$ be a positive integer such that $kx=0$ for all $x\in F$. Then $lky\in X_{0}$. Since $X_{0}$ is connected, the map $x\mapsto lkx$ is a surjective endomorphism of $X_{0}$. We find $q\in X_{0}$ such that $lky=lkq$, and define a virtual 1-cocycle $c_{1}:\Lambda\rightarrow X$ by $c_{1}=c_{y}-c_{q}$. Then for any $\gamma\in\Lambda$,

[TABLE]

Hence the image of $c_{1}$ is contained in $H=\{x\in X:lkx=0\}$. Since $H$ is closed, $\beta$-invariant, and zero-dimensional; from the second condition we conclude that $c_{1}$ is a virtual coboundary. As $c_{1}$ is cohomologous to $c_{y}$, this implies that $c_{y}$ is also a virtual coboundary. Applying Proposition 2.1 we deduce that the coset $y+X$ contains a periodic point $y_{1}$. This contradicts the fact that $y\notin X$, and shows that $Y=X$, i.e., the set of $\beta$-periodic points is dense in $Y$. $\Box$

We now construct an example of a Noetherian action of a finitely generated residually finite group that does not have a dense set of periodic points. Let $H$ denote the group of all functions from ${\mathbb{Z}}$ to ${\mathbb{Z}}$ with finite support, equipped with pointwise addition. We define an action of ${\mathbb{Z}}$ on $H$ by $n\cdot f(i)=f(i+n).$ Let $\Gamma$ denote the semi-direct product of ${\mathbb{Z}}$ and $H$ defined by

[TABLE]

It is easy to see that $\Gamma$ is torsion free. For $k\in{\mathbb{Z}}$, we define $f^{k}\in H$ by $f^{k}(i)=0$ if $k\neq i$ and $f^{k}(i)=1$ if $k=i$. Then for any $k\in{\mathbb{Z}}$, $(0,1)(f^{k+1},0)=(f^{k},0)(0,1)$, i.e., $(0,1)(f^{k+1},0)((0,1)^{-1}=(f^{k},0)$. As $\{(0,f^{k}):k\in{\mathbb{Z}}\}$ generates $H$ as a ${\mathbb{Z}}$-module, this shows that $\Gamma$ is generated by $(0,1)$ and $(f^{0},0)$. In particular, $\Gamma$ is finitely generated. For $k\geq 2$ let $\Gamma_{k}\subset\Gamma$ denote the set of all elements $(f,n)$ such that $n=0$ (mod $k$) and

[TABLE]

It is easy to see that for each $k$, $\Gamma_{k}$ is a normal subgroup of $\Gamma$ and $\Gamma/\Gamma_{k}$ is finite. Moreover for each non-zero $(f,n)\in\Gamma$ there exists $k$ such that $(f,n)\notin\Gamma_{k}$. Hence $\Gamma$ is residually finite.

Let $X$ denote the compact abelian group $({\mathbb{Z}}/2{\mathbb{Z}})^{{\mathbb{Z}}}$, equipped with the product topology and pointwise addition, and let $S:X\rightarrow X$ denote the shift map defined by $S(x)(i)=x(i+1)$. We define an algebraic $\Gamma$-action $\alpha$ on $X$ by $\alpha(f,n)(x)=S^{n}(x)$. It is easy to see that the shift action of $\mathbb{Z}$ on $X$ is Noetherian. Since $\alpha|_{H}$ is trivial and the action of $\Gamma/H\cong\mathbb{Z}$ induced by $\alpha$ is the shift action on $X$, we deduce that $\alpha$ is also Noetherian.

Let $\pi:H\rightarrow X$ denote the homomorphism defined by $\pi(f)(i)=f(i)\ (mod\ 2)$. We define $c:\Gamma\rightarrow X$ by $c(f,n)=\pi(f)$. Then $c((f_{1},n_{1})(f_{2},n_{2}))=\pi(f_{1}+n_{1}\cdot f_{2})=c(f_{1},n_{1})+S^{n_{1}}(\pi(f_{2}))$. Since

[TABLE]

this shows that $c$ is a 1-cocycle of $\alpha$. We pick any $x\in X$ and define $c^{{}^{\prime}}:\Gamma\rightarrow X$ by $c^{{}^{\prime}}(\gamma)=\alpha(\gamma)(x)-x$. Let $\Lambda$ be an arbitrary finite index subgroup of $\Gamma$. Since $\alpha(\gamma)=I$ for all $\gamma\in H$, it follows that $c^{{}^{\prime}}(H\cap\Lambda)=\{0\}$. On the other hand, the restriction of $c$ to $H$ is a homomorphism with infinite image. As $H\cap\Lambda$ is a finite index subgroup of $H$, we deduce that $c(H\cap\Lambda)$ is also infinite. Hence $c|_{\Lambda}\neq c^{{}^{\prime}}|_{\Lambda}$. As $\Lambda$ and $x$ are arbitrary, we conclude that $c$ is not a virtual coboundary. We now apply Proposition 2.3. Let $\beta$ denote the action corresponding to the cocycle $c$. Since $\alpha$ is Noetherian, so is $\beta$. Applying Proposition 2.1 we conclude that $\beta$ does not have a dense set of periodic points.

3. Density of periodic orbits

In this section we concentrate on the case when $\Gamma$ is polycyclic-by-finite. A countable group $\Gamma$ is polycyclic if there exists a decreasing sequence of subgroups

[TABLE]

such that for each $i$, $\Gamma_{i}$ is a normal subgroup of $\Gamma_{i+1}$ and $\Gamma_{i+1}/\Gamma_{i}$ is cyclic. Any such series is called a polycyclic series of $\Gamma$. A group $\Gamma$ is called polycyclic-by-finite if it contains a finite index subgroup that is polycyclic. If $\Gamma$ is polycyclic-by-finite then every subgroup of $\Gamma$ is finitely generated, and ${\mathbb{Z}}(\Gamma)$ is a Noetherian ring.

Let $\Gamma$ be a polycyclic-by-finite group, $\Gamma_{0}\subset\Gamma$ be a polycyclic subgroup of finite index and $\Gamma_{0}=\Gamma_{n}\supset\Gamma_{n-1}\supset\cdots\supset\Gamma_{0}=\{0\}$ be a polycyclic series of $\Gamma_{0}$. Then the number of $i$’s such that $\Gamma_{i+1}/\Gamma_{i}$ is infinite cyclic is independent of $\Gamma_{0}$ and the polycyclic series. This number is known as the *Hirsch number * of $\Gamma$. The following proposition summarizes some basic properties of this invariant (see [10]).

Proposition 3.1**.**

Let $\Gamma$ be a polycyclic-by-finite group.

1. (1)

$h(\Gamma)=0$* if and only if $\Gamma$ is finite.* 2. (2)

If $\Gamma_{1}$ is a subgroup of $\Gamma$ then $h(\Gamma_{1})\leq h(\Gamma)$. 3. (3)

If $\Gamma_{1}\subset\Gamma$ is normal then $h(\Gamma)=h(\Gamma_{1})+h(\Gamma/\Gamma_{1})$.

We note that the previous proposition applies to finitely generated virtually nilpotent groups since they are polycyclic-by-finite. In the proof of Theorem 1.1 we will also use the following result about polycyclic-by-finite groups (see [8]):

Proposition 3.2**.**

Let $\Gamma$ be a polycyclic-by-finite group and let $M$ be a simple $\mathbb{Z}[\Gamma]$-module. Then $M$ is finite.

Our next lemma is a direct consequence of this result.

Lemma 3.3**.**

Let $\Gamma$ be a polycyclic-by-finite group and let $(X,\alpha)$ be a Noetherian action of $\Gamma$ on a non-trivial compact abelian group $X$. Then $X$ admits a non-zero periodic point.

Proof. Let ${\mathcal{A}}$ be the collection of all proper ${\mathbb{Z}}(\Gamma)$-submodules of ${\widehat{X}}$. Since $\alpha$ is Noetherian, ${\mathcal{A}}$ contains a maximal element $M$. It is easy to see that $N={\widehat{X}}/M$ is a simple ${\mathbb{Z}}(\Gamma)$-module. By the previous proposition $N$ is finite. Let $i:{\widehat{N}}\rightarrow X$ denote the dual of the projection map $\pi:{\widehat{X}}\rightarrow N$. Since $\pi$ is surjective, $i$ is injective. Hence $i({\widehat{N}})$ is a non-trivial $\alpha$-invariant finite subgroup of $X$. Clearly every non-zero point of $i({\widehat{N}})$ is periodic. $\Box$

Lemma 3.4**.**

Let $\Gamma$ be a polycyclic-by-finite group and let $\alpha$ be a Noetherian action of $\Gamma$ on a zero-dimensional group $X$ such that $H^{1}_{v}(X,\alpha)\neq\{0\}$. If ${\rm Ker}(\alpha)=\{\gamma:\alpha(\gamma)=I\}$ is infinite then there exists a polycyclic-by-finite group $\Gamma^{{}^{\prime}}$ with $h(\Gamma^{{}^{\prime}})<h(\Gamma)$, and a Noetherian $\Gamma^{{}^{\prime}}$-action $\alpha^{{}^{\prime}}$ on $X$ such that $H^{1}_{v}(X,\alpha^{{}^{\prime}})\neq\{0\}$.

Proof. Let $\Lambda\subset\Gamma$ be a finite index subgroup, and let $c:\Lambda\rightarrow X$ be a virtual 1-cocycle of $\alpha$ that is not a virtual coboundary. Replacing $\Gamma$ by $\Lambda$ if necessary, we may assume that $\Lambda=\Gamma$. Since ${\rm Ker}(\alpha)$ acts trivially on $X$, it follows that the restriction of $c$ to ${\rm Ker}(\alpha)$ is a homomorphism. Since $\Gamma$ is polycyclic-by-finite, the subgroup ${\rm Ker}(\alpha)$ is finitely generated. This implies that $c({\rm Ker}(\alpha))$ is also finitely generated. By Proposition 2.2 there exists $k\geq 2$ such that $kx=0$ for all $x\in X$. Hence $c({\rm Ker}(\alpha))$ is a finite subgroup of $X$. We define $M=\{\gamma\in{\rm Ker}(\alpha):c(\gamma)=0\}$. Since ${\rm Ker}(\alpha)$ is finitely generated and $M\subset{\rm Ker}(\alpha)$ has finite index, $M$ contains a finite index characteristic subgroup of ${\rm Ker}(\alpha)$, i.e., there exists a finite index subgroup $N\subset M$ such that $\theta(N)=N$ for all automorphism $\theta$ of ${\rm Ker}(\alpha)$. Since for any $g\in\Gamma$ the map $\gamma\mapsto g\gamma g^{-1}$ is an automorphism of ${\rm Ker}(\alpha)$, we deduce that $N$ is a normal subgroup of $\Gamma$. We define $\Gamma^{{}^{\prime}}=\Gamma/N$. As $N$ is a finite index subgroup of ${\rm Ker}(\alpha)$ and ${\rm Ker}(\alpha)$ is infinite, $N$ is also infinite. In particular $h(N)>0$. Hence

[TABLE]

As $\alpha(\gamma)=I$ for all $\gamma\in N$, $\alpha$ induces a Noetherian $\Gamma^{{}^{\prime}}$-action $\alpha^{{}^{\prime}}$ on $X$. We define $c^{{}^{\prime}}:\Gamma^{{}^{\prime}}\rightarrow X$ by $c^{{}^{\prime}}(\gamma N)=c(\gamma)$. We note that for any $\gamma\in\Gamma$ and $n\in N$, $c(\gamma n)=c(\gamma)+\alpha(\gamma)(c(n))=c(\gamma)$. Hence $c^{{}^{\prime}}$ is a well defined virtual 1-cocycle of $\alpha^{{}^{\prime}}$. Suppose there exist a finite index subgroup $\Lambda^{{}^{\prime}}\subset\Gamma^{{}^{\prime}}$ and $x\in X$ such that $c^{{}^{\prime}}(a)=\alpha^{{}^{\prime}}(a)(x)-x$ for all $a\in\Lambda^{{}^{\prime}}$. We define $\Lambda_{1}=\pi^{-1}(\Lambda^{{}^{\prime}})$, where $\pi:\Gamma\rightarrow\Gamma^{{}^{\prime}}$ is the projection map. Then for any $\gamma\in\Lambda_{1}$,

[TABLE]

Since this contradicts the fact that $c$ is not a virtual coboundary, we conclude that $c^{{}^{\prime}}$ is also not a virtual coboundary. Hence $H^{1}_{v}(X,\alpha^{{}^{\prime}})\neq 0$.$\Box$

Our next result shows that for Notherian actions of virtually nilpotent groups on zero-dimensional compact abelian groups, the virtual first cohomology group vanishes. In view of Theorem 2.5 and Lemma 3.3 this completes the proof of Theorem 1.1.

Theorem 3.5**.**

Let $\Gamma$ be a finitely generated virtually nilpotent group, and let $(X,\alpha)$ be a Noetherian action of $\Gamma$ on a zero-dimensional group $X$. Then $H^{1}_{v}(\alpha)=\{0\}$

*Proof. * Let $\Lambda\subset\Gamma$ be a finite index subgroup, and let $c:\Lambda\rightarrow X$ be a virtual 1-cocycle of $\alpha$. Suppose $\Gamma_{0}\subset\Gamma$ is a nilpotent subgroup of finite index. Then $c$ is equivalent to $c|_{\Gamma_{0}\cap\Lambda}$. Hence, to show that $c$ is a virtual coboundary, without loss of generality we may assume that $\Lambda\subset\Gamma_{0}$.

We will use induction on $h(\Lambda)$, the Hirsch number of $\Lambda$. If $h(\Lambda))=0$ then $\Lambda$ is finite, and hence $c$ is a virtual coboundary. Suppose $h(\Lambda)\geq 1$. Then $\Lambda$ is a finitely generated infinite nilpotent group. Hence the center of $\Lambda$ is also finitely generated and infinite ([4], Proposition 2.8). We choose an element $\gamma_{0}$ in the center of $\Lambda$ that has infinite order. For $m\geq 1$, we define $K_{m}=(\alpha(\gamma_{0}^{m})-I)(X)$. Since $\alpha(\gamma_{0})$ commutes with $\alpha(\gamma)$ for all $\gamma\in\Gamma$, it follows that each $K_{m}$ is a closed $\alpha$-invariant subgroup. As $\alpha$ is Noetherian, the collection $\{K_{m}:m\geq 1\}$ has a minimal element $K$. Clearly $K$ is of the form $(\alpha(\gamma_{1})-I)(X)$, where $\gamma_{1}=\gamma_{0}^{n}$ for some $n\geq 1$. Let $\beta$ denote the induced algebraic $\Gamma$-action on the quotient $X/K$. If $P$ denotes the projection map from $X$ to $X/K$ then $P\circ c$ is virtual 1-cocycle of the action $\beta$. We note that the $\beta(\gamma)=I$ for all $\gamma$ in the infinite cyclic subgroup generated by $\gamma_{1}$. By Lemma 3.4 and the induction hypothesis we deduce that $P\circ c$ is a virtual coboundary of $\beta$. We find a finite index subgroup $\Lambda_{1}\subset\Lambda$ and $p\in X/K$ such that

[TABLE]

We choose $q\in X$ such that $P(q)=p$. Let $c_{1}:\Lambda_{1}\rightarrow X$ denote the virtual 1-cocycle defined by $c_{1}=c-c_{q}$. Since $P$ is an equivariant map it follows that $P\circ c_{1}=0$, i.e., the image of $c_{1}$ is contained in $K=(\alpha(\gamma_{1})-I)(X)$. We choose $l\geq 1$ such that $\gamma_{1}^{l}\in\Lambda_{1}$. From the minimality of $K$ we deduce that $K=(\alpha(\gamma_{2})-I)(X)$, where $\gamma_{2}=\gamma_{1}^{l}$. We find $x\in X$ such that $\alpha(\gamma_{2})(x)-x=c_{1}(\gamma_{2})$. Let $c_{2}:\Lambda_{1}\rightarrow X$ be the virtual 1-cocycle defined by $c_{2}(\gamma)=c_{1}(\gamma)-c_{x}(\gamma)$. Then $c_{2}$ is cohomologous to $c_{1}$ and $c_{2}(\gamma_{2})=0$. Let $F\subset X$ denote the set of points that are fixed by $\alpha(\gamma_{2})$. Since $\alpha(\gamma_{2})$ commutes with $\alpha(\gamma)$ for all $\gamma$, $F$ is a $\alpha$-invariant closed subgroup. We note that for any $\gamma\in\Lambda_{1}$, $c_{2}(\gamma_{2}\gamma)=c_{2}(\gamma_{2})+\alpha(\gamma_{2})(c_{2}(\gamma))=\alpha(\gamma_{2})(c_{2}(\gamma))$. As $\gamma_{2}$ lies in the center of $\Gamma$ we also have,

[TABLE]

This shows that for all $\gamma\in\Lambda_{1}$ the element $c_{2}(\gamma)$ lies in $F$. Let $\alpha_{F}$ denote the restriction of $\alpha$ to $F$. Then $c_{2}$ can be viewed as a virtual 1-cocycle of $\alpha_{F}$. Since $\alpha(\gamma)|_{F}=I$ for all $\gamma$ in the infinite cyclic group generated by $\gamma_{2}$, from the previous lemma and the induction hypothesis we conclude that $c_{2}$ is a virtual coboundary. Since $c$ is cohomologous to $c_{2}$ this completes the proof. $\Box$

We conclude this paper with the following question :

Question 3.6**.**

Does Theorem 1.1 hold if $\Gamma$ is polycyclic-by-finite ?

In view of Theorem 2.5 and Lemma 3.3 this is true if and only if Theorem 3.5 holds for polycyclic-by-finite groups.