Sofic Groups Research Articles

Entropy on modules over the group ring of a sofic group

We partially generalize Peters' formula on modules over the group ring ${\mathbb F} \Gamma$ for a given finite field ${\mathbb F}$ and a sofic group $\Gamma$. It is also discussed that how the values of entropy are related to the zero divisor conjecture.

Sofic boundaries of groups and coarse geometry of sofic approximations

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups. Along the way, we bring to light a coarse geometric viewpoint on ultralimits of a sequence of finite graphs first exposed by Ján Špakula and Rufus Willett, as well as proving some bridging results concerning measure structures on topological groupoid Morita equivalences that will be of interest to groupoid specialists.

Dynamical correspondences of $L^2$-Betti numbers

We investigate dynamical analogues of the L^2 -Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we show that the L^2 -Betti numbers exactly measure the failure of addition formula for dynamical invariants.

An asymptotic equipartition property for measures on model spaces

Let G G be a sofic group, and let Σ = ( σ n ) n ≥ 1 \Sigma = (\sigma _n)_{n\geq 1} be a sofic approximation to it. For a probability-preserving G G -system, a variant of the sofic entropy relative to Σ \Sigma has recently been defined in terms of sequences of measures on its model spaces that ‘converge’ to the system in a certain sense. Here we prove that, in order to study this notion, one may restrict attention to those sequences that have the asymptotic equipartition property. This may be seen as a relative of the Shannon–McMillan theorem in the sofic setting. We also give some first applications of this result, including a new formula for the sofic entropy of a ( G × H ) (G\times H) -system obtained by co-induction from a G G -system, where H H is any other infinite sofic group.

Metric topological groups: their metric approximation and metric ultraproducts

We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every separable topological group with left-invariant metric. In particular, there is a countable sequence of finite groups with left-invariant metric such that every finite subset of an arbitrary topological group with left-invariant metric may be approximated by all but finitely many of them. We compare our results with related concepts such as sofic groups, hyperlinear groups and weakly sofic groups.

Two special subgroups of the universal sofic group

We define a subgroup of the universal sofic group, obtained as the normalizer of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each sofic representation can be conjugated inside this subgroup.

Some remarks on finitarily approximable groups

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of Doucha. Referring to a problem of Zilber, we show that a the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of Turing. Finally, we solve a conjecture of Pillay by proving that the identity component of a compactification of a pseudofinite group must be abelian as well. All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal.

Soficity and hyperlinearity for metric groups

We define sofic, weakly sofic, linear sofic and hyperlinear metric groups and discuss relationship among these classes.

Local and doubly empirical convergence and the entropy of algebraic actions of sofic groups

Let$G$be a sofic group and$X$a compact group with$G\curvearrowright X$by automorphisms. Using (and reformulating) the notion of local and doubly empirical convergence developed by Austin, we show that in many cases the topological and the measure-theoretic entropy with respect to the Haar measure of$G\curvearrowright X$agree. Our method of proof recovers all known examples. Moreover, the proofs are direct and do not go through explicitly computing the measure-theoretic or topological entropy.

Naive entropy of dynamical systems

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.

Metric mean dimension for algebraic actions of Sofic groups

We prove that if Γ \Gamma is a sofic group and A A is a finitely generated Z ( Γ ) \mathbb {Z}(\Gamma ) -module, then the metric mean dimension of Γ ↷ A ^ , \Gamma \curvearrowright \widehat {A}, in the sense of Hanfeng Li, is equal to the von Neumann-Lück rank of A . A. This partially extends the results of Hanfeng Li and Bingbing Liang from the case of amenable groups to the case of sofic groups. Additionally we show that the mean dimension of Γ ↷ A ^ \Gamma \curvearrowright \widehat {A} is the von Neumann-Lück rank of A A if A A is finitely presented and Γ \Gamma is residually finite. It turns out that our approach naturally leads to a notion of p p -metric mean dimension, which is in between mean dimension and the usual metric mean dimension. This can be seen as an obstruction to the equality of mean dimension and metric mean dimension. While we cannot decide if mean dimension is the same as metric mean dimension for algebraic actions, we show in the metric case that for all p p the p p -metric mean dimension coincides with the von Neumann-Lück rank of the dual module.

Some closure results for 𝒞-approximable groups

We investigate closure results for C-approximable groups, for certain classes C of groups with invariant length functions. In particular we prove, each time for certain (but not necessarily the same) classes C that (i) the direct product of two C-approximable groups is C-approximable; (ii) the restricted standard wreath product G wr H is C-approximable when G is C-approximable and H is residually finite; and (iii) a group G$ with normal subgroup N is C-approximable when N is C-approximable and G/N is amenable. Our direct product result is valid for LEF, weakly sofic and hyperlinear groups, as well as for all groups that are approximable by finite groups equipped with commutator-contractive invariant length functions (considered in \cite{Thom}). Our wreath product result is valid for weakly sofic groups, and we prove it separately for sofic groups. We note that this last result has recently been generalised by Hayes and Sale, who prove that the restricted standard wreath product of any two sofic groups is sofic. Our result on extensions by amenable groups is valid for weakly sofic groups, and was proved by Elek and Szabo for sofic groups N.

Approximations of groups, characterizations of sofic groups, and equations over groups

We give new characterizations of sofic groups:–A group G is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups.–A group G is sofic if and only if any system of equations solvable in all alternating groups is solvable over G. The last characterization allows to express soficity of an existentially closed group by ∀∃-sentences.

A non-LEA sofic group

We describe elementary examples of finitely presented sofic groups which are not residually amenable (and thus not initially subamenable or LEA, for short). We ask if an amalgam of two amenable groups over a finite subgroup is residually amenable and answer this positively for some special cases, including countable locally finite groups, residually nilpotent groups and others.

The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

Abstract Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

Positive Sofic Entropy Implies Finite Stabilizer

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 measures. This extends the 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.

The Entropy of Gibbs Measures on Sofic Groups

We show that for every local potential on a sofic group there exists a shift-invariant Gibbs measure. Under some conditions we show that the sofic entropy of the corresponding shift action does not depend on a sofic approximation. Bibliography: 12 titles.

Linear sofic groups and algebras

We introduce and systematically study linear sofic groups and linear sofic algebras. This generalizes amenable and LEF groups and algebras. We prove that a group is linear sofic if and only if its group algebra is linear sofic. We show that linear soficity for groups is a priori weaker than soficity but stronger than weak soficity. We also provide an alternative proof of a result of Elek and Szabo which states that sofic groups satisfy Kaplansky’s direct finiteness conjecture.

Fuglede–Kadison Determinants and Sofic Entropy

We relate Fuglede-Kadison determinants to entropy of algebraic actions of sofic groups in essentially complete generality. This generalizes recent results of Hanfeng Li and Andreas Thom from the amenable case to the sofic case, as well as results of David Kerr, Hanfeng Li, and Lewis Bowen in the residually finite case. Moreover, the proof given is the first calculation of entropy of algebraic actions to avoid a nontrivial determinant approximation. We apply our results to a problem of Christopher Deninger, generalizing results of Hanfeng Li and David Kerr. Finally, we show that in many cases, finiteness of topological entropy for algebraic actions of sofic groups is equivalent to having metric mean dimension zero.

POLISH MODELS AND SOFIC ENTROPY

We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if$\unicode[STIX]{x1D6E4}$is a nonamenable group and$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.