Positive sofic entropy implies finite stabilizer

Positive sofic entropy implies finite stabilizer

Tom Meyerovitch
Abstract.

We prove that for a measure preserving action of a sofic group with positive sofic entropy, the stabilizer is finite on a set of positive measure. This extends results of Weiss and Seward for amenable groups and free groups, respectively. It follows that the action of a sofic group on its subgroups by inner automorphisms has zero topological sofic entropy, and that a faithful action that has completely positive sofic entropy must be free.

1. Introduction

The last decade brought a number of important developments in dynamics of non-amenable group actions. Among these we note the various extensions of classical entropy theory. For actions of free groups, L. Bowen introduced a numerical invariant known as -invariant entropy [6]. Some time later Bowen defined new invariants for actions of sofic groups, called sofic entropy [4]. Kerr and Li further developed sofic entropy theory and also adapted it to groups actions on topological spaces by homeomorphisms [8]. The classical mean-entropy for amenable groups and Bowen’s -invariant both turned out to be special cases of sofic entropy [3, 9].

The study of non-free measure-preserving group actions is another fruitful and active trend in dynamics. These are closely related to the notion of invariant random subgroups: Namely, a probability measure on the space of subgroups whose law is invariant under conjugation. Any such law can be realized as the law of the stabilizer for a random point for some probability preserving action [1]. In this note we prove the following:

Theorem 1.1.

Suppose is an action of a countable sofic group that has positive sofic entropy (with respect to some sofic approximation). Then the set of points in with finite stabilizer has positive measure. In particular, if the action is ergodic, almost every point has finite stabilizer.

The amenable case, Weiss pointed out that the conclusion of Theorem 1.1 as a remark in the last section of his survey paper on actions of amenable groups [13]. To be precise, Weiss stated the amenable case of Corollary 5.1 below.

Another interesting case of Theorem 1.1 for free groups is due to Seward [11]. The result proved in [11] applies to the random sofic approximation. By a non-trivial result of Bowen this coincides with the -invariant for free groups. Sewards’s proof in [11] is based on a specific formula for -entropy, which does not seem to be available for sofic entropy in general. Our proof below proceeds essentially by proving a combinatorial statement about finite objects. In personal communication, Seward informed me of another proof of Theorem 1.1 that is expected to appear in a forthcoming paper of Alpeev and Seward as a byproduct of their study of an entropy theory for general countable groups.

Theorem 1.1 confirms the point of view that the “usual” notions of sofic entropy for sofic groups (or mean-entropy in the amenable case) are not very useful as invariants for non-free actions. A version of sofic entropy for actions with stabilizers was developed by Bowen [5] as a particular instance of a more general framework of entropy theory of sofic groupoids. It seems likely that both the statement of Theorem 1.1 and our proof should have a generalization to sofic class bijective extensions of groupoids. We will not pursue this direction.

Acknowledgments. I thank Yair Glasner, Guy Salomon, Brandon Seward and Benjy Weiss for interesting discussions.

2. Notation and definitions

2.1. Sofic groups

Sofic groups were introduced by Gromov [7] (under a different name) and by Weiss [12] towards the end of the millennium. Sofic groups retain some properties of finite groups. They are a common generalization of amenable and residually finite groups. We include a definition below. There are several other interesting equivalent definitions. There are many good references in the literature for further background, motivation and discussions on sofic groups, for instance [10].

Throughout we will use the notation to indicate that is a finite subset of . For a finite set , let denote the group of permutations over . We will consider maps from a group to . These maps are not necessarily homomorphisms. Given a map , and , we write for the image of under and for the image of under the permutation .

Let and . A map is called an -approximation of if it satisfies the following properties:

(1)

and

(2)

A sofic group is a group that admits an -approximation for any and any .

A symmetric -approximation of is that in addition to (1) and (2) also satisfies

(3)

Standard arguments show that a sofic group admits a symmetric -approximation for any and any , so from now assume our -approximations also satisfy (3).

A sequence of maps is called a sofic approximation for if

is co-finite in , for any and any .

2.2. Sofic entropy

Roughly speaking the sofic entropy of an action is if there are “approximately” “sufficiently distinct good approximations” for the action that “factor through” a finite “approximate action” . Various definitions have been introduced in the literature, that have been shown to lead to an equivalent notion. Most definitions involve some auxiliary structure. Here we follow a recent presentation of sofic entropy by Austin [2]. Ultimately, this presentation is equivalent to Bowen’s original definition and also to definitions given by Kerr and Li.

Let be a probability preserving action on a standard probability space. As explained in [2], by passing to an isomorphic action we can assume without loss of generality that , where is a compact metric space and that the action of is the shift action: , and that is a Borel probability measure on , where the Borel structure is with respect to the product topology.

More specifically, we will assume that is equipped with the metric , where . These assumptions can be made without loss of generality. Indeed, start with an arbitrary (standard) Borel space , choose a countable sequence of Borel subsets so that the smallest -invariant -algebra containing is the Borel -algebra. There is a -equivariant Borel embedding of to defined by

Let denote the push-forward measure of , it follows that the -action on is isomorphic to the -action on . Also note that

(4)

In particular, the diameter of is .

We recall some definitions and notation that Austin introduced in [2]:

Definition 2.1.

Given , and , the pullback name of at , denoted by is defined to be:

(5)

The empirical distribution of with respect to is defined by:

(6)

Given a -neighborhood of , the set of -approximations for the action is given by

In [2] elements of are called “good models”.

The space , if it is non-empty, is considered as a metric space with respect to the following metric

Given a compact metric space and we denote by the maximal cardinality of a -separated set in , and by the minimal number of -balls of radius needed to cover . Let us recall a couple of classical relations between these quantities. Because distinct -separated points cannot be in the same -ball the following holds:

Consider a maximal -separated set . The collection of -balls with centers in covers . Thus:

Definition 2.2.

Let be a sofic approximation of , with . The -entropy (or sofic entropy with respect to ) of is defined by:

(7)

where the infimum is over weak- neighborhoods of in .If for all large ’s, define .

The key fact is that the quantity does not depend on the topological model or on the choice of metric , and is thus an invariant for the action , with respect to isomorphism in the class of probability preserving actions.

Remark 2.3.

We recall a slight generalization of -entropy: A random sofic approximation is where so that the conditions (1) and (2) hold “on average” with respect to for any and , if is large enough.

In this case -entropy is defined by

(8)

For the special case is a free group on generators and is chosen uniformly among the homomorphisms from to the group of permutations of , Bowen proved that -entropy coincides with the so called -invariant [3].

Our proof of Theorem 1.1 applies directly with no changes to random sofic approximations, in particular to -entropy.

2.3. Stabilizers and the space of subgroups

Let denote the space of subgroups of . The space comes with a compact topology, inherited from the product topology on . The group acts on by inner automorphisms. Now let be an action of on a standard Borel space . For let

(9)

The map is Borel and -equivariant.

The following fact about the map appears implicitly for instance in [13]:

Lemma 2.4.

Let be an ergodic action of a countable group. If the action has finite stabilizers, the map induces a finite factor .

Proof.

Suppose is finite on a set of positive measure. By ergodicity on a set of full measure. Since there are only countably many finite subgroups, the measure must be purely atomic. To finish the proof note that a purely atomic invariant probability measure must be supported on a single finite orbit, if it is ergodic. ∎

Here is a quick corollary of Theorem 1.1 that concerns the action :

Corollary 2.5.

Let be an infinite sofic group and a sofic approximation sequence. The topological -entropy of the action by conjugation is zero.

Proof.

The variational principle for -entropy states that the topological -entropy of an action is equal to the supremum of the measure-theoretic -entropy over all -invariant measures [8]. It thus suffices to prove that any -invariant measure on has zero -entropy. By Theorem 1.1, it is enough to show that the set is null. Indeed, for any , , because any subgroup is contained in its normalizer. Thus groups with finite stabilizer must be finite, so is a countable set. Suppose . It follows that is purely atomic. As in Lemma 2.4, each ergodic component of must be supported on finite set. An action of an infinite group on a finite set can not have finite stabilizers. This shows that . ∎

3. Sampling from finite graphs

In this section we prove an auxiliary result on finite labeled graphs.

We begin with some terminology:

Definition 3.1.

A finite , simple and directed graph is a pair where is a finite set and (we allow self-loops but no parallel edges).

  • The out-degree and in-degree of are given by

  • is -regular if at most vertices have out-degree less than , and all vertices have in-degree at most .

  • A set is -dominating if the number of vertices in so that is at most .

  • A -Bernoulli set for is a random subset of such that for each the probability that is , independently of the other vertices.

Lemma 3.2.

Fix any . Suppose satisfy

(10)

For any -regular graph with , a -Bernoulli subset is -dominating and has size at most with probability at least .

Proof.

Suppose (10) holds. Let be a graph satisfying the assumptions in the statement of the lemma, and let be -Bernoulli.

For , let be number of edges with . The random variable is Binomial . Let

It follows that

Thus

For , the random variables and are independent, unless there is a common vertex which both and . Because the maximal in-degree is at most , each can account for at most such pairs, so there are at most pairs which are not independent. Also note that for every so . It follows that

By Chebyshev’s inequality, the probability that is not -dominating is at most

Also and , so again by Chebyshev’s inequality

It follows that with probability at least , is -dominating and . ∎

4. Proof of Theorem 1.1

Suppose is infinite -almost-surely. Our goal is to prove that the sofic entropy of the -action is non-positive with respect to any sofic approximation (in the case of a deterministic approximation sequence this means it is either or ). By a direct inspection of the definition of sofic entropy in (7), our goal is to show that for any given there exists a neighborhood of so that for any sufficiently good approximation ,

We will show that we can choose the neighborhood to be of the form (see Definition 4.2 below), for some parameters and .

Definition 4.1.

(Approximate stabilizer) For and and let

(11)
Definition 4.2.

Let and . Define

to be the set of probability measures satisfying the following conditions:

(12)
(13)
Lemma 4.3.

If is not an integer power of , the set is open.

Proof.

Suppose is not an integer power of . By (4) it follows that if and only if . So for every ,

It follows that for any and the set is a clopen set: It is both open and closed in .

Because and are both finite,

and

are also clopen in . So the indicator functions are continuous. Now

so is an open set. ∎

We now specify how to choose the parameters , , and are chosen according to :

  • Choose so that

    (14)
  • Choose , so that is not an integer power of and a finite subset depending on and on the measure so that

    (15)

    Where is defined in Definition 4.1 above. The is possible by Lemma 4.4 below. We also require

    (16)
  • Choose depending on and big enough so that

    (17)

    It is clear that the left hand side in both expressions tends to as , so such choice of is indeed possible.

  • Choose a finite subset depending on on and on the measure so that and

    (18)

    We prove the existence of such a set in Lemma 4.5 below.

  • Choose another finite subset so that , and

    (19)
  • Choose big enough so that

    (20)
  • Choose to be a -approximation of .

Lemma 4.4.

For any measure and (15) holds for some and sufficiently small .

Proof.

Note that

(21)

Also, whenever and . So by -additivity of ,

It follows that (15) holds for some and sufficiently small . ∎

Lemma 4.5.

Under the assumption that is infinite -almost-surely, for every and there exists so that and (18) holds.

Proof.

Because -a.e, it follows that for any ,

Note that

So as in the proof of Lemma 4.4 using -additivity of , it follows that (18) holds for some . Furthermore, we can assume that by further increasing . ∎

Lemma 4.6.

For , and as above, .

Proof.

Because , it follows that

So by (15) it follows that (13) also holds with replaced by . Using (15) and (18) and the condition we see that (12) holds with replaced by .

Thus .

The following lemma shows that approximate stabilizers behave well under conjugation:

Lemma 4.7.

If and satisfies

(22)

then

(23)
Proof.

Suppose (22) holds.

Choose any . By (22),

(24)

Now choose any . For any we have so we can substitute instead of in (24) to obtain

Now and

So we have

This means that .

We conclude that (22) implies (23). ∎

Definition 4.8.

Call good for if the following conditions are satisfied:

(25)
(26)

and

(27)

Otherwise, say that is bad for .

Lemma 4.9.

Let with . Then there exists a set and a function with the following properties:

  1. .

  2. .

where is defined by

(28)
Proof.

Consider the directed graph with

Because the approximation is symmetric, the maximal in-degree in is at most . Let denote the set of ’s for which the mapping is injective on . Because is a sufficiently good approximation of it follows that , so is -regular.

By Lemma 3.2, a -Bernoulli set is -dominating set and has size less than with probability at least . To see that Lemma 3.2 applies, we used the left inequality in (17) (keeping in mind that ), and (20) to deduce that (10) is satisfied with and replaced with and . In this case satisfies . For choose randomly as follows: Whenever the set is non-empty, choose uniformly at random from . If let be chosen uniformly at random from . We see that if is -dominating is satisfied.

To conclude the proof we will show that is satisfied with probability at least .

For and denote:

(29)

Because of , it follows that for any , all but an -fraction of the ’s are good so

(30)

Now let denote the indicator of the event “ is bad for

is a random variable, because is a random function.

Note that

(31)

It follows that for ,

Because it follows that

(32)

Because is a permutation:

So from (32) and (30) we get that for every

Averaging over :

Using Markov inequality, it follows that

So holds with probability at least . ∎

Given a metric space and a finite set , the following “hamming-like” metric is defined on :

(33)

We also have the following “uniform” metric on , where is a finite set:

(34)

We will use the following relatively estimate:

Lemma 4.10.

For any finite set and we have

Proof.

If is such that and then the union of -balls in with centers in covers . It follows that

The claim now follows by the following standard and easily verified facts:

We record the following Lemma (see [2, Lemma ], and recall that we use a left-action):

Lemma 4.11.

Suppose is good for and then

Proof.

Because is good for it follows that

so for every we have