Positive sofic entropy implies finite stabilizer

We prove that for a measure preserving action of a sofic group with positive sofic entropy, the set of points with finite stabilizer have 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 a faithful action with completely positive sofic entropy must be free.


Notation and definitions
To avoid some numerical constants, we use the notation F 1 (x 1 , . . . , x n ) ≪ F 2 (x 1 , . . . , x n ) to indicate that there exists an explicit constant K > 0, independent of any of the parameters x 1 , . . . , x n so that F 1 (x 1 , . . . , x n ) < KF 2 (x 1 , . . . , x n ) for any choice of Throughout G is a countable group which acts on a standard probability space (X, B, µ) by probability preserving bijections. For a finite or countable partition α ⊂ B of X, and F ⋐ G we denote For infinite A ⊂ G we denote A · α the σ-algebra generated by the sets A slight abuse of notation: We will sometimes identify a partition α of X with the function x → α(x) sending an element of X to it's partition cell. Sofic groups: Let F ⋐ G and δ > 0. A map σ : G → S d is called an (F, δ)-sofic approximation if for any g, h ∈ F Following Weiss [7] (also Gromov [3], under a different name), a group G is sofic if there exists an (F, δ)-sofic approximation for any F ⋐ G and δ > 0. A sequence is called a sofic-approximation sequence if for any δ > 0 and F ⋐ G there exists i 0 ∈ N such that σ i is an (F, δ)-sofic approximation for all i > i 0 .
From now we assume G is a sofic group and (σ i ) ∞ i=1 is a fixed sofic approximation. Sofic entropy: Suppose α ⊂ B is a finite partition of X which is dynamically generating, namely G · α = B mod µ. The existence of a finite dynamically generating partition is not automatic, it's a condition we assume to simplify the presentation.
Bowen [2] introduced a notion of "sofic entropy" for sofic groups. The sofic entropy of (X, B, µ, σ) with respect to a sofic approximation sequence Σ = (σ i ) ∞ i=1 is given by: where α * Map µ (α, F, δ, σ i ) := {α • φ : φ ∈ Map µ (α, F, δ, σ i )} If there exist F ⊂ G and δ > 0 so that Map µ (α, F, δ, σ i ) = ∅ for all large i we define h µ (X, B, Σ) = −∞. Bowen showed in [2] that quantity h µ (X, B, Σ) does not depend on the choice of finite generating partition α. The are other ways to define sofic entropy, for instance as in [4]. These lead to an equivalent notion of sofic entropy in case there is a finite dynamically generating partition. Note that the finite set F ⋐ G above plays the role of two different parameters in appearing in the definition of sofic entropy in [4]: It is both the set of elements g for which the action of Σ mimics the action of G, and the collection of "observables" with respect to which P d • φ −1 approximates µ.

Positive Entropy implies finite stabilizer
Lemma 2.1 (Small ǫ-dominating random subsets in high degree graphs ). For any ǫ > 0 there exists k 0 sufficiently large such that for all k > k 0 and any M > 0 there exists N = N (k, M ) such that any directed graph with at least N vertices and maximal in degree at most M such that all but ǫn vertices have out degree at least k, a Bernoulli-randomly chosen set of intensity 1 √ k is 2ǫ-dominating with probability at least 1 − ǫ.
Proof. (Sketch) Let G = (V, E) be a graph as above. Choose a random set C by selecting each vertex independently with probability 1 Thus the expected number of vertices with no edge pointing at some w ∈ C at most ǫ For v, w ∈ V . the random variables n(v) and n(w) are independent, unless there is a common vertex u which has an incoming edge from both u and w. Because the maximal in-degree is at most M , each u ∈ V can account for at most M 2 such pairs, so there are at most M 2 |V | pairs which are not independent, so using second moment estimate, Chebyshev's inequality give that the probability that more than 3 The following simple lemma is a twist on an observation known as the "Mass Transport Principle". In this case, as in many other applications it is almost trivial, yet surprisingly useful. See for instance [5] for a more general context of the mass transport principle: B, µ) is an ergodic G-action with positive sofic entropy. Then the stabilizer is finite µ-almost-surely.
Proof. Note that by ergodicity |stab(x)| is equal to a constant off a µ-null set, and in particular it is either finite with probability 1 or infinite with probability 1. Suppose stab(x) is infinite µ-almost-surely. Let α be a dynamically generating partition.
Choose M ≫ ǫ −2 . From the assumption that stab(x) is infinite µ-almost-surely, it follows that there exists F 0 ⋐ G sufficiently big so that From the assumption that α is a dynamically generating partition it follows that we can find F 1 ⋐ G sufficiently big so that for any g ∈ F 0 By making F 0 and then F 1 bigger, we assume without loss of generality that F 0 and F 1 are symmetric F 0 = F −1 0 and F 1 = F −1 1 . We also assume 1 ∈ F 0 ⊂ F 1 . Now choose an even bigger F 2 ⋐ G, namely it should satisfy Choose a set Map ⊂ Map µ (α, F, δ, σ). so that α * Map = α * Map µ (α, F, δ, σ) and |Map| = |α * Map|.
Consider the graph G F2 whose vertex set is [d] such the edge (i, j) exists if and only if σ g (i) = j for some g ∈ F 2 .
We will describe a procedure for choosing a random function τ : [d] → G. First, choose a random subset C ⊂ [d] so that i ∈ C with probability 1 √ |F2| independently for i = 1, . . . , d. Given the subset C as above, for each i ∈ [d], τ (i) will be chosen uniformly from the set If the set N i above is empty, τ (i) is chosen uniformly from F 2 . We denote by ν the probability measure on the measure space on which τ and C are defined. Also define σ τ : Because σ : G → S d is an (F, δ)-sofic approximation, all but δd vertices in the graph G F2 have degree at least (1 − δ)|F 2 |, and all vertices have in-degree at most |F 2 |. It follows using Lemma 2.1 that a randomly chosen set C ⊂ [d] as above will be ǫ-dominating with high probability. It follows that with high probability with respect to ν we get C and τ so that The set stab φ (i) ⊂ G should be viewed as a "guess" for elements which are in the stabilizer of φ(i), using "local observations".

4
For any φ ∈ Map, because (φ, σ) is a sufficiently good sofic model for G (X, B, µ), it follows that for any g ∈ F 0 Using (1), for any φ ∈ Map we have: good for φ if the following conditions are satisfied: Because σ is an (F, δ)-sofic approximation the third property fails only on a set of size ≪ ǫd. If φ ∈ Map the last property also fails only in for a set of size ≪ ǫd. We denote by : Ψ φ : [d] → R the indicator function of bad points: (4) and (5) it follows that for any φ ∈ Map The last equability above is due to the fact that Thus, Using the Mass Transport Principle (Lemma 2.2), we thus have: Using Markov inequality, it follows that for some explicit constant K > 0 In particular there exists τ : and a set Map 0 ⊂ Map so that |Map 0 | > 1 2 such that for any φ ∈ Map 0 We now fix C and τ satisfying the above. For j ∈ [d] we define stab φ (j) ⋐ G as follows: The set stab j (φ) ⊂ F should again be viewed as a "guess" for elements which are in the stabilizer of φ(j), based on local information obtained by "sampling only around points in C". Note that for any j ∈ [d] we have We say that i ∈ [d] is exceptional for φ and τ if one of the following conditions hold: (1) | stab φ (i)| < M (2) There exists g ∈ stab φ (i) so that α(φ(i)) = α(φ(σ g (i)).
For ψ ∈ Map − 0 we denote Given φ ∈ Map 0 , we define another graph G = G φ , whose vertices are [d] and there is an edge from i to j if and only if j = σ g (i) for some g ∈ stab φ (i). Notice that the graph G φ only depends on φ| CF 2 1 , so it makes sense to write G ψ for ψ ∈ Map − 0 . Let C 1 ⊂ [d] be a random subset so that i ∈ C 1 with probability 1 √ M independently for i ∈ [d]. It follows using Lemma 2.1 that with positive probability a randomly chosen set C 1 ⊂ [d] as above is ǫ-dominating in G ψ and that In particular, for any ψ ∈ Map − 0 there exists C 1 as above, which we denote by C ψ . For ψ ∈ Map − 0 , let κ ψ (i) := max{j ∈ C ψ : (i, j) is an edge in G ψ }, where κ ψ is undefined if {j ∈ C ψ : (i, j) is an edge in G ψ } = ∅. Note that if i is not exceptional for φ and τ then κ(i) is defined and α • φ)(i) = (α • φ)(κ ψ (i)). It follows that for every φ ∈ Map ψ 0 we have: Thus any φ 1 , φ 2 ∈ Map ψ 0 differ only on a set of size ≪ ǫd. So for some constant K > 0 we have for any ψ ∈ Map − 0 , so using Stirling's approximation, for ǫ ∈ (0, 1 2 ): log Map ψ 0 ≪ (log |α|ǫ − ǫ log ǫ) d, Also note that Note that We conclude: For any ǫ > 0, there exists δ > 0 and F ⋐ G so that for any (F, δ)-sofic approximation σ : G → S d log Map µ (α, F, δ, σ) ≪ (log |α|ǫ + ǫ log ǫ) d, So the action G (X, B, µ) does not have positive sofic entropy.