Locally approximating groups of homeomorphisms of manifolds

TL;DR

This paper studies groups of homeomorphisms of manifolds that can be locally approximated, revealing their deep logical structure, model-theoretic properties, and how they determine the topology and geometry of the underlying manifolds.

Contribution

It proves that locally approximating groups of homeomorphisms interpret first order arithmetic and encode subgroup membership, leading to new insights into their model theory and topological rigidity.

Findings

Locally approximating groups interpret first order arithmetic.

Finitely generated locally approximating groups are often prime models.

The dimension and topology of the manifold are determined by the group's elementary theory.

Abstract

Let $M$ be a compact, connected manifold of positive dimension and let $\mathcal G\leq\textrm{Homeo}(M)$ be \emph{locally approximating} in the sense that for all open $U\subseteq M$ compactly contained in a single Euclidean chart of $M$, the subgroup $\mathcal G[U]$ consisting of elements of $\mathcal G$ supported in $U$ is dense in the full group of homeomorphisms supported in $U$. We prove that $\mathcal G$ interprets first order arithmetic, as well as a first order predicate that encodes membership in finitely generated subgroups of $\mathcal G$. As a consequence, we show that if $\mathcal G$ is not finitely generated, then no group elementarily equivalent to $\mathcal G$ can be finitely generated. We show that many finitely generated locally approximating groups of homeomorphisms $\mathcal G$ of a manifold are prime models of their theories, and give conditions that guarantee any finitely presented group $G$ that is elementarily equivalent to $\mathcal G$ is isomorphic to $\mathcal G$. We thus recover some results of Lasserre about the model theory of Thompson's groups $F$ and $T$. Finally, we obtain several action rigidity result for locally approximating groups of homeomorphisms. If $\mathcal G$ acts in a locally approximating way on a compact, connected manifold $M$ then the dimension of $M$ is uniquely determined by the elementary equivalence class of $\mathcal G$. Moreover, if $\dim M\leq 3$ then $M$ is uniquely determined up to homeomorphism. In for general closed smooth manifolds, the homotopy type of $M$ is uniquely determined. In this way, we obtain a generalization of a well-known result of Rubin.