Submission date: 24 November 2020

Abstract:

We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical core J associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of J, modulo certain infinitesimal automorphisms, is a locally compact group G. The automorphism groups of models of the theory are related with G, not in general via a homomorphism, but by a quasi-homomorphism, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of SL_n(ℝ) or SL_n(ℚ_p).

Mathematics Subject Classification:

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Full text arXiv 2011.12009: pdf, ps.