Power quotients of surface groups and mapping class groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914032411213824 |
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| author | Coulon, Rémi Sisto, Alessandro Wilton, Henry |
| author_facet | Coulon, Rémi Sisto, Alessandro Wilton, Henry |
| contents | Let $Γ$ be the fundamental group of a closed, orientable, hyperbolic surface $S$. The $n$-power quotient, $Γ(n)$, is the quotient of $Γ$ by the $n$th powers of simple closed curves. We prove an analogue of the Dehn--Nielsen--Baer theorem for suitable large values of $n$: the outer automorphism group of $Γ(n)$ is isomorphic to the quotient of the extended mapping class group of $S$ by $n$th powers of Dehn twists. There is also a corresponding description of the automorphism group as the quotient of the extended mapping class group of the corresponding once-punctured surface, and we relate these groups via a Birman-type exact sequence. Along the way, and as consequences, we prove structural properties of $Γ(n)$ for suitable large values of $n$, including: $Γ(n)$ is virtually torsion-free, acylindrically hyperbolic, infinitely presented, with solvable word problem and finite asymptotic dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13701 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Power quotients of surface groups and mapping class groups Coulon, Rémi Sisto, Alessandro Wilton, Henry Group Theory Geometric Topology Let $Γ$ be the fundamental group of a closed, orientable, hyperbolic surface $S$. The $n$-power quotient, $Γ(n)$, is the quotient of $Γ$ by the $n$th powers of simple closed curves. We prove an analogue of the Dehn--Nielsen--Baer theorem for suitable large values of $n$: the outer automorphism group of $Γ(n)$ is isomorphic to the quotient of the extended mapping class group of $S$ by $n$th powers of Dehn twists. There is also a corresponding description of the automorphism group as the quotient of the extended mapping class group of the corresponding once-punctured surface, and we relate these groups via a Birman-type exact sequence. Along the way, and as consequences, we prove structural properties of $Γ(n)$ for suitable large values of $n$, including: $Γ(n)$ is virtually torsion-free, acylindrically hyperbolic, infinitely presented, with solvable word problem and finite asymptotic dimension. |
| title | Power quotients of surface groups and mapping class groups |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2507.13701 |