Locally approximating groups of homeomorphisms of manifolds

• Main Authors: Koberda, Thomas, González, J. de la Nuez
• Format: Journal Article
• Language: English
• Published: 21-10-2024
• Subjects: Mathematics - Geometric Topology; Mathematics - Group Theory; Mathematics - Logic
• DOI: 10.48550/arxiv.2410.16108

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.