Metric Approximations of Wreath Products

Hayes, Ben; Sale, Andrew W.

doi:10.5802/aif.3166

Cited by 11 publications

(32 citation statements)

References 19 publications

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“…There are three different proofs of (3) in [DKP14,Pȃu11,ES11]. Soficity of graph products is studied in [CHR14] and wreath products in [HS16,HS18].…”

Section: Which Groups Are Sofic?mentioning

confidence: 99%

Examples in the entropy theory of countable group actions

Bowen¹

2019

Ergod. Th. Dynam. Sys.

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Kolmogorov-Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

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“…There are three different proofs of (3) in [DKP14,Pȃu11,ES11]. Soficity of graph products is studied in [CHR14] and wreath products in [HS16,HS18].…”

Section: Which Groups Are Sofic?mentioning

confidence: 99%

Examples in the entropy theory of countable group actions

Bowen¹

2019

Ergod. Th. Dynam. Sys.

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“…In a sense, extensions we consider in this paper are opposite to amenable-by-sofic groups, and nothing is known about metric approximations of such groups. For instance, it follows from [11,Corollary 3.8] (see also [10,Corollary 2], where the authors focus on restricted regular wreath products) that if F is a finitely generated free group with normal subgroup S, and F/S is sofic, then also the group F/S ′ is sofic, where S ′ denotes the derived subgroup of S.…”

Section: 4mentioning

confidence: 99%

Unrestricted wreath products and sofic groups

Arzhantseva

Berlai

Finn-Sell

et al. 2019

Int. J. Algebra Comput.

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We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argument, to hyperlinearby-amenable groups.2010 Mathematics Subject Classification. 20F65, 03C25. Key words and phrases. Wreath product, amenable, sofic, and hyperlinear groups. 1 2 G. ARZHANTSEVA, F. BERLAI, M. FINN-SELL, AND L. GLEBSKY Theorem B. Let G be a hyperlinear group and H be an amenable group. Then the unrestricted wreath product G ≀≀ H is hyperlinear.As hyperlinearity is preserved under taking subgroups, using the Kaloujnine-Krasner Theorem, we immediately obtain Corollary 1.1. Let G be a group with a hyperlinear, normal subgroup N G such that the quotient G/N is amenable. Then G is hyperlinear.This fact is perhaps known to specialists but it had not yet appeared in the literature explicitly.In the final section we indicate that the results are optimal for such an approach: it does not immediately extend neither to sofic-by-{residually amenable} nor to {weakly sofic}-by-amenable group extensions.

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“…These are all sofic: graph products (e.g. free or direct products) of sofic groups [4], amalgamated free products or HNN extensions of sofic groups over amenable groups [5,9,22], wreath products of sofic groups [15], groups with finite index sofic subgroups, limits of sofic groups in the space of marked groups (but not all finitely generated sofic groups are limits of amenable groups [6]), locally sofic groups (e.g. groups locally embeddable into sofic groups or direct limits of sofic groups).…”

Section: Soficitymentioning

confidence: 99%

Soficity and variations on Higman’s group

2019

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A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag-Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways.We construct variations on Higman's group replacing the Baumslag-Solitar group by other groups G. We give an elementary condition on G, enjoyed for example by Z ≀ Z and the integral Heisenberg group, under which the resulting group is sofic.We then use soficity to deduce that there exist permutations of Z/nZ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.

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Metric Approximations of Wreath Products

Cited by 11 publications

References 19 publications

Examples in the entropy theory of countable group actions

Examples in the entropy theory of countable group actions

Unrestricted wreath products and sofic groups

Soficity and variations on Higman’s group

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