Asymptotic Approximations of Finitely Generated Groups

Arzhantseva, Goulnara

doi:10.1007/978-3-319-05488-9_2

Cited by 10 publications

(11 citation statements)

References 17 publications

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“…The notion of F-approximable group for various choices of F captures many classes of groups that are currently a subject of intensive study. We mention only main examples below, for more details about these wide classes of groups and a broad range of applications see [37,36,17,1,3,15]. The reader is also invited to analyze the F-profile and F-dimension, given her/his favorite family F. Moreover, note that our quantifying of metric approximations extends immediately to quantifying of constraint metric approximations [4], using suitable constraint metric profiles.…”

Section: Classes Of F-approximable Groupsmentioning

confidence: 99%

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Quantifying metric approximations of discrete groups

Arzhantseva¹,

Cherix²

2020

Preprint

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We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group G by a family of groups F = {(Gα, dα, kα, εα)}α∈I, where each group Gα is equipped with a bi-invariant metric dα and a dimension kα, for strictly positive real numbers εα such that infα εα > 0. Through the notion of a residually amenable profile that we introduce, our approach generalizes classical isoperimetric (aka Følner) profiles of amenable groups and recently introduced functions quantifying residually finite groups. Our viewpoint is much more general and covers hyperlinear and sofic approximations as well as many other metric approximations such as weakly sofic, weakly hyperlinear, and linear sofic approximations.

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Section: Classes Of F-approximable Groupsmentioning

confidence: 99%

“…A group G is hyperlinear if and only if G embeds into a metric ultrapower of Ū [36,Corollary 4.3]. In [1], the first author observed that all Gromov hyperbolic groups G are residually finite (respectively, LE-F f in , LEA, etc.) if and only if G embeds into Ū .…”

Section: 41mentioning

confidence: 99%

Quantifying metric approximations of discrete groups

Arzhantseva¹,

Cherix²

2020

Preprint

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“…In the particular case when the finite generating set X n does not depend on n one obtains the notion of C-approximation as above. It is showed in [4,Theorem 11] that several important classes of finitely generated groups, including all hyperbolic groups and one-relator groups, are asymptotically residually finite, i.e. have the asymptotic C-approximation property where C is the class of residually finite groups endowed with the trivial length function.…”

Section: Ii9 Other Metric Approximations and Higman's Groupmentioning

confidence: 99%

Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

Capraro

Lupini

2015

Lecture Notes in Mathematics

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PrefaceAnalogy is one of the most effective techniques of human reasoning: When we face new problems we compare them with simpler and already known ones, in the attempt to use what we know about the latter ones to solve the former ones. This strategy is particularly common in Mathematics, which offers several examples of abstract and seemingly intractable objects: Subsets of the plane can be enormously complicated but, as soon as they can be approximated by rectangles, then they can be measured; Uniformly finite metric spaces can be difficult to describe and understand but, as soon as they can be approximated by Hilbert spaces, then they can be proved to satisfy the coarse Novikov's and Baum-Connes's conjectures.These notes deal with two particular instances of such a strategy: Sofic and hyperlinear groups are in fact the countable discrete groups that can be approximated in a suitable sense by finite symmetric groups and groups of unitary matrices. These notions, introduced by Gromov and Rȃdulescu, respectively, at the end of the 1990s, turned out to be very deep and fruitful, and stimulated in the last 15 years an impressive amount of research touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and more recently even quantum information theory. Several long-standing conjectures that are still open for arbitrary groups were settled in the case of sofic or hyperlinear groups. These achievements aroused the interest of an increasing number of researchers into some fundamental questions about the nature of these approximation properties. Many of such problems are to this day still open such as, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? A similar pattern can be found in the study of II 1 factors. In this case the famous conjecture due to Connes (commonly known as Connes' embedding conjecture) that any II 1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs in the understanding of II 1 factors, and stands out today as one of the major open problems in the field.The aim of this monograph is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and Connes' embedding conjecture. These notions, as well as the proofs of many results, are here presented in the framework of model theory for metric structures. We believe that this point of view, even though rarely explicitly adopted in the literature, can contribute to a better understanding iii iv of the ideas therein, as well as provide additional tools to attack many remaining open problems. The presentation is nonetheless self-contained and accessible to any student or researcher with a graduate-level mathematical background. In particular no specific knowledge of logic or model theory is required.Chapter 1 presents the conjectures and open problems that will serve as common thread and motivation for the rest of the survey: Connes' em...

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“…in Group Theory (e.g. [ESS18], [HS18], [CGLT17], [BL18], [Arz14], [AP15], [Atk18], [MM01], [BLT18], [BM18], [GR09]).…”

Section: Introductionmentioning

confidence: 99%

Almost commuting matrices, cohomology, and dimension

Enders¹,

Shulman²

2019

Preprint

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It is an old problem to investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative C * -algebras C(X) are matricially semiprojective. Extending the works of Davidson, Eilers-Loring-Pedersen, Lin and Voiculescu on almost commuting matrices, we identify the precise dimensional and cohomological restrictions for finite-dimensional spaces X and thus obtain a complete characterization: C(X) is matricially semiprojective if and only if dim(X) ≤ 2 and H 2 (X; Q) = 0. We give several applications to lifting problems for commutative C * -algebras, in particular to liftings from the Calkin algebra and to l-closed C * -algebras in the sense of Blackadar.

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Asymptotic Approximations of Finitely Generated Groups

Cited by 10 publications

References 17 publications

Quantifying metric approximations of discrete groups

Quantifying metric approximations of discrete groups

Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

Almost commuting matrices, cohomology, and dimension

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