Locally approximating groups of homeomorphisms of manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916475459076096 |
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| author | Koberda, Thomas González, J. de la Nuez |
| author_facet | Koberda, Thomas González, J. de la Nuez |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16108 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locally approximating groups of homeomorphisms of manifolds Koberda, Thomas González, J. de la Nuez Group Theory Geometric Topology Logic 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. |
| title | Locally approximating groups of homeomorphisms of manifolds |
| topic | Group Theory Geometric Topology Logic |
| url | https://arxiv.org/abs/2410.16108 |