Sofic boundaries of groups and coarse geometry of sofic approximations
| Autor: | Martin Finn-Sell, Vadim Alekseev |
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| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
Boundary (topology) Group Theory (math.GR) 01 natural sciences Measure (mathematics) symbols.namesake Mathematics - Metric Geometry 0103 physical sciences FOS: Mathematics Discrete Mathematics and Combinatorics 0101 mathematics Operator Algebras (math.OA) Equivalence (measure theory) Mathematics Sequence 20L05 20F65 46L55 Group (mathematics) Mathematics::Operator Algebras 010102 general mathematics Hilbert space Mathematics - Operator Algebras Metric Geometry (math.MG) symbols 010307 mathematical physics Geometry and Topology Finitely generated group Mathematics - Group Theory Group theory |
| DOI: | 10.48550/arxiv.1608.02242 |
| Popis: | Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups. Along the way, we bring to light a coarse geometric view point on ultralimits of a sequence of finite graphs first exposed by Jan \v{S}pakula and Rufus Willett, as well as proving some bridging results concerning measure structures on topological groupoid Morita equivalences that will be of interest to groupoid specialists. Comment: 33 pages, to appear in Groups Geom. Dyn |
| Databáze: | OpenAIRE |
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