Hyperlinear and Sofic Groups: A Brief Guide

Pestov, Vladimir

doi:10.2178/bsl/1231081461

Cited by 183 publications

(198 citation statements)

References 61 publications

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“…A note on nonresidually solvable hyperlinear one-relator groups This paper concerns the sofic property discussed in the survey [Pestov 2008]. Particularly, we address Question 4.10 in that paper: the problem of Nate Brown asking whether or not every one-relator group is sofic.…”

Section: Applicationmentioning

confidence: 99%

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2010

Involve

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Section: Applicationmentioning

confidence: 99%

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2010

Involve

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“…by standard amplification argument [Pe08]. Indeed, the direct product of finite groups is obviously finite and one can define a bi-invariant distance on the direct product as the sum of the bi-invariant metrics on the factors.…”

Section: The Original Definition In [Glri08] Is Algebraic and Uses A mentioning

confidence: 99%

“…We mainly use the later technique due to simplicity in writing. For a careful introduction to the subject, including ultraproducts terminology, see [Pe08,PeKw09].…”

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confidence: 99%

Linear sofic groups and algebras

Arzhantseva

Paunescu

2016

Trans. Amer. Math. Soc.

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ABSTRACT. 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. These group properties can be stated in elementary algebraic terms, as approximation properties, or in the language of ultraproducts, as the existence of an embedding in

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“…(If there is some sequence of finite approximating graphs, the group is called sofic. It is not known if there are non-sofic groups [Pes08], [AlLy07].) It is also unclear what the analogue of property (3) should be.…”

Section: Percolation Renormalization and Scale-invariant Tilingsmentioning

confidence: 99%

Scale-invariant groups

Nekrashevych¹,

Pete²

2011

Groups Geom. Dyn.

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Abstract. Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F o Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS.1; m/; the affine groups A Ë Z d , for any A Ä GL.Z; d /. However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.Mathematics Subject Classification (2010). 20E08, 20F18, 05B45, 60B99, 60K35.

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Hyperlinear and Sofic Groups: A Brief Guide

Cited by 183 publications

References 61 publications

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Linear sofic groups and algebras

Scale-invariant groups

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