Beyond the Lascar Group

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 {\em core} $\mathcal{J}$ associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $\mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $\mathcal{G}$. The automorphism groups of models of the theory are related with $\mathcal{G}$, not in general via a homomorphism, but by a {\em 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 $S{L}_n(\mathbb{R})$ or $S{L}_n({\mathbb{Q}}_p)$.

v3: a few local improvements; an editing error above 5.28 fixed; a more general treatment of minimal commensurability class of subgroups containing a given commensurability class of approximate subgroups (5.10)

References in corpus (2)

• Topological dynamics and the complexity of strong types
• Approximate invariance for ergodic actions of amenable groups

Cited by in corpus (1)

• Symbolic substitution systems beyond abelian groups