2024/10/21 by Thomas Koberda, Koberda, Thomas, J. de la Nuez González +1
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Logic (math.LO) #Mathematical Dynamics and Fractals #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2410.16108
openalex publication_date 2024/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a compact, connected manifold of positive dimension and let \mathcal G≤\textrmHomeo(M) be locally approximating in the sense that for all open U⊆ 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≤ 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.