Isometry groups of infinite-genus hyperbolic surfaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866914707652214784 |
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| author | Aougab, Tarik Patel, Priyam Vlamis, Nicholas G. |
| author_facet | Aougab, Tarik Patel, Priyam Vlamis, Nicholas G. |
| contents | Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_01982 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Isometry groups of infinite-genus hyperbolic surfaces Aougab, Tarik Patel, Priyam Vlamis, Nicholas G. Geometric Topology Complex Variables Group Theory Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups. |
| title | Isometry groups of infinite-genus hyperbolic surfaces |
| topic | Geometric Topology Complex Variables Group Theory |
| url | https://arxiv.org/abs/2007.01982 |