Sofic boundaries of groups and coarse geometry of sofic approximations

Abstract

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 Špakula 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.

arXiv comment33 pages, to appear in Groups Geom. Dyn

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Cited by in corpus (5)

• Coarse fundamental groups and box spaces
• Group approximation in Cayley topology and coarse geometry, Part II: Fibered coarse embeddings
• Coarse amenability at infinity
• Non-standard limits of graphs and some orbit equivalence invariants
• Dynamic characterizations of quasi-isometry, and applications to cohomology