Classification of aut-fixed subgroups in free-abelian times surface groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915378186158080 |
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| author | Lei, Jialin Wang, Peng Zhang, Qiang |
| author_facet | Lei, Jialin Wang, Peng Zhang, Qiang |
| contents | In this paper, we are concerned with the direct product $G=π_1(Σ)\times \Z^k$ for $Σ$ a compact orientable surface with negative Euler characteristic, and give a complete classification of its fixed subgroups of automorphisms. As a corollary, we show that $G$ contains, up to isomorphism, infinitely many fixed subgroups of automorphisms if and only if $k\geq 2$, which is a contrast to that of hyperbolic groups. As an application on Nielsen fixed point theory, we provide a family of aspherical manifolds without Jiang's Bound Index Property. Moreover, we also give some results on the fixed subgroups of the direct product $H\times \Z^k$ for $H$ a non-elementary torsion-free hyperbolic group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_13540 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification of aut-fixed subgroups in free-abelian times surface groups Lei, Jialin Wang, Peng Zhang, Qiang Group Theory Geometric Topology 20F65, 20F34, 57M07 In this paper, we are concerned with the direct product $G=π_1(Σ)\times \Z^k$ for $Σ$ a compact orientable surface with negative Euler characteristic, and give a complete classification of its fixed subgroups of automorphisms. As a corollary, we show that $G$ contains, up to isomorphism, infinitely many fixed subgroups of automorphisms if and only if $k\geq 2$, which is a contrast to that of hyperbolic groups. As an application on Nielsen fixed point theory, we provide a family of aspherical manifolds without Jiang's Bound Index Property. Moreover, we also give some results on the fixed subgroups of the direct product $H\times \Z^k$ for $H$ a non-elementary torsion-free hyperbolic group. |
| title | Classification of aut-fixed subgroups in free-abelian times surface groups |
| topic | Group Theory Geometric Topology 20F65, 20F34, 57M07 |
| url | https://arxiv.org/abs/2309.13540 |