Detecting Free Group Automorphisms via Virtual Homology Representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912216439062528 |
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| author | Yüksel, Emre |
| author_facet | Yüksel, Emre |
| contents | Let $F_n= F\langle x_1,...,x_n\rangle$ denote the free group of rank $n\ge 2$ and let $\mathrm{End}(F_n)$ be the endomorphism monoid of $F_n$. We show that automorphisms of $F_n$ are detected via the $\mathrm{End}(F_n)$-action on the first integral homology of finite characteristic covers of the wedge of $n\ge 2$ circles $R_n$. This gives a homological characterization of homotopy equivalences of $R_n$ that we utilize to show that $\mathrm{End}(F_n)$ is asymptotically linear. We extend these results by showing that the $\mathrm{Out}(F_n)$-action on the homology of iterated covers of a punctured surface $Σ_g^b$ of the same homotopy type as $R_n$ detects homeomorphisms of $Σ_g^b$ in homotopy classes of homotopy equivalences of $R_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13803 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Detecting Free Group Automorphisms via Virtual Homology Representations Yüksel, Emre Geometric Topology 57M10, 57M60, 57K20 Let $F_n= F\langle x_1,...,x_n\rangle$ denote the free group of rank $n\ge 2$ and let $\mathrm{End}(F_n)$ be the endomorphism monoid of $F_n$. We show that automorphisms of $F_n$ are detected via the $\mathrm{End}(F_n)$-action on the first integral homology of finite characteristic covers of the wedge of $n\ge 2$ circles $R_n$. This gives a homological characterization of homotopy equivalences of $R_n$ that we utilize to show that $\mathrm{End}(F_n)$ is asymptotically linear. We extend these results by showing that the $\mathrm{Out}(F_n)$-action on the homology of iterated covers of a punctured surface $Σ_g^b$ of the same homotopy type as $R_n$ detects homeomorphisms of $Σ_g^b$ in homotopy classes of homotopy equivalences of $R_n$. |
| title | Detecting Free Group Automorphisms via Virtual Homology Representations |
| topic | Geometric Topology 57M10, 57M60, 57K20 |
| url | https://arxiv.org/abs/2501.13803 |