An overview of Benjamini-Schramm convergence in group theory and dynamics

An overview of Benjamini-Schramm convergence in group theory and dynamics

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Lewis Bowen, University of Texas, Austin & Princeton University
Fine Hall 110

When studying an infinite geometric object or graph it is natural to want a "good" finite or bounded model for the sake of computations. But what does "good" mean here? This notion is formalized by Benjamini-Schramm convergence: "good" means that locally the finite object looks like the infinite one, except for a small density of singularities. While this notion appears rather weak, it is surprisingly useful.  Indeed, there are many problems in transitive graphs, countable groups and stationary processes (for example) that can be solved only by viewing the underlying structure as a limit of finite models. A small sampling  includes: classification of Bernoulli shifts over sofic groups, direct finiteness for group rings, computation of L2 betti numbers and other spectral invariants. This point of view also leads to new invariants which in turn lead to nonembeddability and nonisomorphism results.This talk will be a broad overview intended for a general audience. No background or specialized knowledge will be assumed.

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