Following the mechanisms, where the coarse geometric properties of a space with bounded geometry can induce properties on the related coarse (boundary) groupoid and vice versa, we prove that a sequence of bounded degree graphs being hyperfinite is equivalent to the equivalence relation induced by the coarse boundary groupoid associated to this sequence being hyperfinite. Even more, we introduce a coarse and weaker notion of Property A in a sequence of graphs, called Property A on average, that also turns out to be equivalent to the hyperfiniteness of a sequence of bounded degree graphs. Furthermore, we show that if the coarse boundary groupoid is topologically a-T-menable, then the related sequence of bounded degree graphs is asymptotically coarsely embeddable into a Hilbert space. In the measurable case, we also have the asymptotic coarse embeddability of the sequence of graphs after discarding small subgraphs along the sequence and looking at this new sequence of graphs with the induced length metric of original graph.
Afterwards those result are applicable to sofic groups. When we take the sequence of graphs to be a sofic approximation of an amenable discrete finitely generated sofic group, we know that this sequence is hyperfinte, has property A on average and property almost-A. If the group is a-T-menable then the sequence of graphs is weakly asymptotically coarsely embeddable into a Hilbert space.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:76706 |
| Date | 23 November 2021 |
| Creators | Businhani Biz, Leonardo |
| Contributors | Alekseev, Vadim, Thom, Andreas, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | info:eu-repo/grantAgreement/Deutsche Forschungsgemeinschaft/SPP 2026/339872158//Asymptotic geometry of sofic groups and manifolds |
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