Sofic boundaries of groups and coarse geometry of sofic approximations

Алексеев, В. А.; Finn-Sell, Martin

doi:10.48550/arxiv.1608.02242

Cited by 6 publications

(15 citation statements)

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“…This paper continues the previous work by first named author and Martin Finn-Sell from [AFS16], relating coarse geometry of the graph spaces obtained from sofic approximations of a group Γ and analytic properties of the group, using the coarse boundary groupoid to connect them. There, it was proven that coarse-geometric properties of the sofic approximation like property A or asymptotic coarse embeddability into a Hilbert space imply amenability resp.…”

Section: Introductionsupporting

confidence: 62%

“…The next results can also be founded in the Remark 3.12 of [AFS16] with an operator algebra approach and in the case where X is the space of graphs of a sofic approximation.…”

Section: Measured Ultraproductsmentioning

confidence: 77%

“…In the same way we did for the Stone-Čech compatification of N, one can see a point in ∂βX as a limit of a sequence along a non-principal ultrafilter ω ∈ ∂βN. Based on that, an important geometric relation between them was proved in [AFS16] and written here for completeness: Proposition 3.17 ([AFS16, Proposition 2.14]). Let ∂G(X) be the boundary groupoid, where X is a space of finite bounded degree graphs.…”

Section: 3mentioning

confidence: 84%

“…In this section we give an overview of some results that link properties of the group (amenability and a-T-menability) with the geometric properties on the sequence of graphs (hyperfiniteness, property A and embeddability into a Hilbert space). The results presented here are taken from [AFS16], [Ele06], [FS14], [Roe03] and [Wil06].…”

Section: 2mentioning

confidence: 99%

“…In this paper we systematically investigate both amenability and a-T-menability of the coarse boundary groupoid ∂G(X) attached to a sequence of bounded degree graphs {X i } i∈N . We make use of the idea from [AFS16] that the coarse boundary groupoid can usefully be considered both as topological and measured groupoid, reflecting the difference between more "rigid' coarse geometric properties and "geometry up to negligible subsets", usually encountered in the context of sofic approximations.…”

Section: Introductionmentioning

confidence: 99%

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Sofic boundaries and a-T-menability

Алексеев¹,

Biz²

2021

Preprint

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We undertake a systematic study of the approximation properties of the topological and measurable versions of the coarse boundary groupoid associated to a sequence of finite graphs of bounded degree. On the topological side, we prove that asymptotic coarse embeddability of the graph sequence into a Hilbert space is equivalent to the coarse boundary groupoid being topologically a-T-menable, thus answering a question by Rufus Willett. On the measure-theoretic side, we prove that measure-theoretic amenability and a-T-menability of the coarse boundary groupoid are related to hyperfiniteness and property almost-A resp. an version of "almost asymptotic embeddability into Hilbert space". These results can be directly applied to spaces of graphs coming from sofic approximations.

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Section: Introductionsupporting

confidence: 62%

“…The next results can also be founded in the Remark 3.12 of [AFS16] with an operator algebra approach and in the case where X is the space of graphs of a sofic approximation.…”

Section: Measured Ultraproductsmentioning

confidence: 77%

Section: 3mentioning

confidence: 84%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sofic boundaries and a-T-menability

Алексеев¹,

Biz²

2021

Preprint

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Combinatorial cost: A coarse setting

Kaiser

2019

Trans. Amer. Math. Soc.

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The main inspiration for this paper is a paper by Elek where he introduces combinatorial cost for graph sequences. We show that having cost equal to 1 and hyperfiniteness are coarse invariants. We also show 'cost−1' for box spaces behaves multiplicatively when taking subgroups. We show that graph sequences coming from Farber sequences of a group have property A if and only if the group is amenable. The same is true for hyperfiniteness. This generalises a theorem by Elek. Furthermore we optimise this result when Farber sequences are replaced by sofic approximations. In doing so we introduce a new concept: property almost-A.

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Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups

Mimura

Sako

2019

J. Topol. Anal.

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The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In this Part I, we prove that a disjoint union has property A of G. Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems of generators that look similar such that one has property A and the other does not admit (fibred) coarse embeddings into any Banach space with non-trivial type (for instance, any uniformly convex Banach space).

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Sofic boundaries of groups and coarse geometry of sofic approximations

Cited by 6 publications

References 0 publications

Sofic boundaries and a-T-menability

Sofic boundaries and a-T-menability

Combinatorial cost: A coarse setting

Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups

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