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Hyperlinear and sofic groups: a brief guide

Vladimir G. Pestov

arXiv:0804.3968v8math.GR

TL;DR

The paper addresses the theory of hyperlinear and sofic groups, whose motivations arise from Connes’ Embedding Conjecture and Gottschalk Surjunctivity. It surveys their concepts, results, and sources through metric ultraproduct characterizations, and discusses open questions alongside established results.

  • Problem

    Hyperlinear and sofic groups arise from major questions concerning type II1 von Neumann factors and Gottschalk Surjunctivity.

  • Method

    The paper offers an introductory survey of the theory, including characterizations through metric ultraproducts of finite-rank groups.

  • Results

    Every sofic group is hyperlinear.

  • Takeaways & Limitations

    The survey connects the two classes through their ultraproduct characterizations while presenting their main theory and open questions.

  • Takeaways & Limitations

    For von Neumann algebras, the analogous ultraproduct statement fails because ultraproducts do not behave well with order completeness.

Abstract

from arXiv · show

Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless. Hyperlinear groups have their origin in Connes' Embedding Conjecture about von Neumann factors of type $II_1$, while sofic groups, introduced by Gromov, are motivated by Gottschalk Surjunctivity Conjecture (can a shift $A^G$ contain a proper isomorphic copy of itself, where $A$ is a finite discrete space and $G$ is a group?). Groups from both classes can be characterized as subgroups of metric ultraproducts of families of certain metric groups (formed in the same way as ultraproducts of Banach spaces): unitary groups of finite rank lead to hyperlinear groups, symmetric groups of finite rank - to sofic groups. We offer an introductory guide to some of the main concepts, results, and sources of the theory, following Connes, Gromov, Benjamin Weiss, Kirchberg, Ozawa, Radulescu, Elek and Szabó, and others, and discuss open questions which are for the time being perhaps more numerous than the results.

1. Introduction

The paper introduces hyperlinear and sofic groups as two recently isolated classes arising from operator algebras and symbolic dynamics. It presents an introductory guide to their theory, including main concepts, results, sources, and open questions.

  • Hyperlinear and sofic groups are discrete, countable classes introduced independently from operator algebras and symbolic dynamics, respectively.
  • Hyperlinear groups originate in Connes’ Embedding Conjecture for type II1 von Neumann factors, while sofic groups are motivated by Gottschalk Surjunctivity.
  • Both classes are characterized using metric ultraproducts of finite-rank unitary or symmetric groups.Finite-rank unitary groups lead to hyperlinear groups, whereas finite-rank symmetric groups lead to sofic groups.
  • The paper provides an introductory guide to major concepts, results, and sources in the theory.
  • It also discusses open questions, which the authors describe as perhaps more numerous than the existing results.

2. Ultraproducts

The section develops ultraproduct constructions for algebraic structures, normed spaces, and metric groups. It emphasizes that bi-invariant metrics are needed for well-defined metric-group ultraproducts and introduces key examples.

  • 2.1. Algebraic ultraproducts.: Algebraic ultraproducts quotient the Cartesian product of structures by threads equal to the identity on an ultrafilter-large set.This construction was introduced by Jerzy Łoś in 1955.
  • 2.2. Ultraproducts of normed spaces.: Normed-space ultraproducts use a bounded finite part of the Cartesian product, quotient infinitesimals, and obtain a Banach space under suitable ultrafilter conditions.For a non-countably-complete ultrafilter, the result is Banach; it is either finite-dimensional when dimensions are ultrafilter-bounded or non-separable otherwise.
  • 2.2. Ultraproducts of normed spaces.: Ultralimits assign each bounded real sequence a number whose ε-neighborhood is attained on an ultrafilter-large set.Every bounded sequence of reals has an ultralimit along a given ultrafilter.
  • 2.3. Ultraproducts of metric groups: first attempt.: Metric-group ultraproducts first imitate the normed-space construction using compatible left-invariant metrics, but the infinitesimal subgroup need not be normal.The subgroup property follows from left invariance, while normality can fail.
  • 2.4. Bi-invariant metrics.: Bi-invariant metrics ensure that infinitesimals form a normal subgroup, making the quotient metric ultraproduct well-defined and complete.With non-countably-complete ultrafilters, these ultraproducts are either non-separable or locally compact.
  • 2.4. Bi-invariant metrics.: Important examples include uniform metrics on measure-preserving transformations, normalized Hamming metrics on finite symmetric groups, and uniform operator metrics on unitary groups.The uniform operator metric on U(H) is bi-invariant, and uniformly bounded metrics make the finite part equal the full Cartesian product in the examples considered.

3. Definitions

Sofic and hyperlinear groups are defined through finite symmetric and unitary approximations, respectively, with every sofic group known to be hyperlinear. Their definitions admit ultrafilter-independent finite approximation forms, while the converse implication and universality questions remain open.

  • Definitions: Sofic groups embed into metric ultraproducts of finite symmetric groups with normalized Hamming distance, while hyperlinear groups embed into ultraproducts of finite unitary groups with normalized Hilbert–Schmidt distance.The two notions are presented side by side through analogous ultraproduct definitions.
  • Relation between the classes: Finite permutation groups embed into unitary groups through permutation matrices, and this yields the theorem that every sofic group is hyperlinear.The normalized Hamming and Hilbert–Schmidt metrics are sufficiently compatible at the metric-ultraproduct level, although they are not even Lipschitz equivalent on each finite symmetric group.
  • Finite approximations: A group is sofic exactly when every finite subset admits arbitrarily accurate finite symmetric-group almost homomorphisms that separate distinct elements by at least 1/4.The almost-homomorphism conditions approximate multiplication and the identity, while the separation condition ensures injectivity in the ultraproduct construction.
  • Finite approximations: Amplification replaces a positive separation δ between two permutations by at least 1 − (1 − δ)^2 = 2δ − δ^2 under diagonal action on [n] × [n].Repeated amplification can raise separation to 1/4 or any real number strictly between 0 and 1, while a sufficiently small initial approximation error preserves the prescribed accuracy.
  • Finite approximations: Hyperlinearity has an analogous finite characterization using maps from finite subsets into U(n), with approximate multiplication, approximate identity, and pairwise Hilbert–Schmidt separation by at least 1/4.The threshold 1/4 is arbitrary and can be replaced by another fixed positive separation requirement below the maximal range.
  • Ultrafilters and open questions: For countable groups, hyperlinearity and soficity essentially do not depend on the chosen nonprincipal ultrafilter, and each property is equivalent to the corresponding property for all finitely generated subgroups.The theory still leaves open whether every hyperlinear group is sofic, whether every group is sofic, and whether every group is hyperlinear.

4. Examples

The section surveys examples of sofic and hyperlinear groups, including residually finite, amenable, initially subamenable, and Baumslag–Solitar groups, while emphasizing unresolved questions and boundaries.

  • Examples: Every residually finite group is sofic, via finite quotients that yield approximations by finite symmetric groups.The construction separates a finite set from the identity in a finite quotient and embeds that quotient into a symmetric group.
  • Examples: Every nonabelian free group is sofic, and hyperlinearity of nonabelian free groups was established independently by Connes and Wassermann.The hyperlinearity result marked the beginning of this research direction.
  • Examples: Every initially subamenable group is sofic; this class includes residually amenable and LEF groups.Initial subamenability means finite pieces of the group can be reproduced inside amenable groups.
  • Examples: The Baumslag–Solitar group ⟨a, b | ab^3a^-1 = b^2⟩ is sofic despite being non-residually finite, because it is residually solvable.Its hyperlinearity was established separately by Radulescu.
  • Open questions: Not every group is initially subamenable, and it remains open whether every sofic group is initially subamenable or every finitely presented group is sofic or hyperlinear.Finitely presented non-amenable simple groups provide examples outside initial subamenability, while their soficity can distinguish the classes.

5. Further criteria of soficity

Soficity admits equivalent descriptions through local finite graph models and essentially free near-actions with finitely additive invariant probability measures.

  • Graph criterion: The graph criterion uses finite edge V-coloured graphs, where V is a finite symmetric generating set and colours record generator-labelled edges.Cayley-graph adjacency is determined by right multiplication by generators in V.
  • Graph criterion: A finitely generated group is sofic exactly when finite edge-coloured graphs locally model its Cayley graph at all but an arbitrarily small fraction of vertices.For every radius N and ε > 0, at least (1 − ε)|Γ| vertices have N-balls isomorphic to the corresponding Cayley-graph ball.
  • Near-actions: A group is sofic exactly when it admits an essentially free near-action on a set carrying a finitely additive probability measure defined on every subset.The near-action assigns almost-everywhere measure-preserving transformations satisfying the group law, with nonidentity elements almost everywhere fixed-point-free.
  • Near-actions: The near-action characterization is constructed from almost homomorphisms into finite symmetric groups and conversely uses paradoxical-decomposition methods.The construction passes through ultrafilter limits of finite almost-actions.
  • Comparison with amenability: Soficity parallels amenability, but the corresponding measure for sofic groups is finitely additive on all subsets rather than merely sigma-additive on Borel sets.Amenable groups are characterized by left-invariant finitely additive measures on all subsets, while standard Cantor actions need only provide Borel measures.

6. Gottschalk Surjunctivity Conjecture

The Gottschalk Surjunctivity Conjecture asks whether shifts contain proper closed invariant copies of themselves; it holds for sofic groups and connects soficity to major open conjectures.

  • Shift systems: A shift A^G is a symbolic dynamical system in which G acts by translations through homeomorphisms on a Cantor space.Here G is countable and A is finite with the discrete topology.
  • Conjecture: Gottschalk’s conjecture states that A^G contains no proper closed G-invariant subspace isomorphic to A^G as a compact G-space.The conjecture quantifies over every countable group G and finite set A.
  • Open question: It remains open whether the conjecture is equivalent to its binary case A = {0, 1}.The text explicitly identifies this as an open question.
  • Sofic groups: Gromov proved Gottschalk Surjunctivity for sofic groups: their shifts contain no proper isomorphic copies of themselves.This is presented as the main advance to date in the section.
  • Consequences: If every group is sofic, both Gottschalk Surjunctivity and the group version of Connes’ Embedding Conjecture follow; a counterexample to surjunctivity would yield a non-sofic group.Such a group need not necessarily be non-hyperlinear.

7. Von Neumann algebras and tracial ultraproducts

The section develops von Neumann algebras and explains why tracial ultraproducts are needed: ordinary norm ultraproducts do not preserve von Neumann structure, while trace-based quotients recover type II1 factors.

  • Von Neumann algebras: Von Neumann algebras are weakly or strongly closed unital C*-subalgebras of B(H), equivalently C*-algebras isometrically isomorphic to dual spaces.B(H) itself is a von Neumann algebra.
  • Factors and traces: A factor has trivial center, and a type II1 factor is finite while infinite-dimensional as a Banach space.Finite factors admit traces; group von Neumann algebras are finite factors under the i.c.c. condition.
  • Factors and traces: There is a unique approximately finite-dimensional von Neumann factor of type II1 up to von Neumann algebra isomorphism, denoted R.AFD means containing an increasing chain of finite-dimensional subalgebras with strongly dense union.
  • Ultraproducts: Normed-space ultraproducts preserve C*-algebras but not von Neumann algebras because ultraproducts can fail order completeness.A bounded increasing sequence of positive elements may lack a least upper bound in the ultraproduct even when one exists in a larger von Neumann algebra.
  • Tracial ultraproducts: Tracial ultraproducts quotient the bounded direct sum by Hilbert–Schmidt-norm infinitesimals while retaining the usual norm for finite elements.The construction is presented for type II1 factors equipped with normalized traces.
  • Tracial ultraproducts: The resulting C*-algebra is a type II1 factor, and its unitary group is the metric ultraproduct of the component unitary groups with normalized Hilbert–Schmidt metrics.The GNS representation supplies a faithful representation, while density identifies the unitary groups involved.

8. Connes’ Embedding Conjecture

The section presents Connes’ Embedding Conjecture for groups and its equivalence with universal hyperlinearity, including a characterization through group von Neumann algebras and metric ultraproducts.

  • Connes’ Embedding Conjecture asks whether every separable type II1 factor embeds into a suitable tracial ultrapower.
  • For every countable group G, the conjecture for groups asks whether V N(G) embeds into the tracial ultrapower Rω.
  • Hyperlinear groups can also be characterized using metric ultrapowers of the unitary group U(R) of the AFD type II1 factor.
  • A countable group G is hyperlinear exactly when V N(G) embeds into Rω.
  • Connes’ Conjecture for Groups is equivalent to the statement that every group is hyperlinear.
  • The section records open questions about the relationship between hyperlinear groups and groups embedding into ultraproducts of matrix-unitary groups, and whether the latter class contains every countable group.

9. Some classes of groups to look at

This section surveys group classes whose relationship to hyperlinearity or soficity remains unresolved, including property (T), word-hyperbolic, Burnside, Haagerup, and exact groups.

  • It remains open whether every infinite simple group with Kazhdan’s property (T) can be hyperlinear or sofic.
  • It remains open whether every word-hyperbolic group is hyperlinear or sofic, alongside the related residual-finiteness question.
  • It remains open whether the free Burnside group of a finite exponent n is sofic.
  • A negative answer to Question 9.4 would imply a non-residually-finite word-hyperbolic group and therefore a negative answer to Question 9.3.
  • It remains open whether every group with the Haagerup property is hyperlinear or sofic.
  • It remains open whether every amenable-at-infinity, equivalently exact, group is hyperlinear or sofic; a positive answer would imply a positive answer to Question 4.10.

10. Equations in groups

The section connects regular equations in groups with finite-group extension results and hyperlinearity, showing how Connes’ Embedding Conjecture relates to a longstanding equation-solving conjecture.

  • Every regular equation in a finite group has a solution in a finite group extending that group.
  • It remains conjectured that every regular equation in a group has a solution in some group extending it.
  • Every regular equation with coefficients in U(n) has a solution in U(n), with the result extending to all compact connected Lie groups and certain systems of equations.
  • Every regular equation with coefficients in a hyperlinear group has a solution in a suitable hyperlinear group extension, obtainable as a metric ultraproduct of finite-rank unitary groups.
  • If Connes’ Embedding Conjecture for groups holds, then the regular-equations conjecture holds; any counterexample to the latter disproves Connes’ conjecture.

11. Varia

The concluding miscellany records consequences of universal soficity, related areas and permanence questions, weak soficity, and set-theoretic observations about the survey’s main problems.

  • The survey is explicitly non-exhaustive and points readers toward material it leaves out.
  • If every group is sofic, then Kaplansky’s Direct Finiteness Conjecture, the Determinant Conjecture, and other conjectures would be settled positively.
  • Sofic groups are linked to stochastic processes in infinite networks, cellular automata, and classification of Bernoulli actions.
  • The section raises whether a group can have a sofic or hyperlinear radical, requiring closure of products of normal sofic or hyperlinear subgroups.
  • Weakly sofic groups are defined using metric ultraproducts of arbitrary finite groups with bi-invariant metrics, and it is conjectured that every group is weakly sofic.
  • Set theory suggests that the survey’s main problems are unlikely to be independent and may not be affected by adding several standard axioms to ZFC.

12. Some reading suggestions

The article recommends several introductory and complementary sources on sofic groups, hyperlinear groups, ultraproducts, and operator algebras. It also presents a Gromov-attributed dichotomy suggesting that counterexamples to Connes’ Embedding Conjecture for Groups should be sought.

  • Weiss’ survey is recommended as a strong introduction to sofic groups, followed by Elek and Szabó’s treatment of both sofic and hyperlinear groups.
  • Pisier’s Section 9.10 offers an enjoyable, largely self-contained discussion of ultraproducts and Connes’ Conjecture.
  • Weaver’s introduction and Ozawa’s survey provide complementary perspectives on C∗-algebras, von Neumann algebras, and the broader theory.
  • A Gromov-attributed dichotomy says that any statement about all countable groups is either trivial or false.
  • Following this dictum, the article argues that researchers should seek counterexamples to Connes’ Embedding Conjecture for Groups unless its proof is unexpectedly simple.
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