Dehn filling Dehn twists
| Autor: | François Dahmani, Mark F. Hagen, Alessandro Sisto |
|---|---|
| Přispěvatelé: | Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Department of Mathematics [ETH Zurich] (D-MATH), Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Dahmani, Francois, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology in Zürich [Zürich] (ETH Zürich) |
| Rok vydání: | 2018 |
| Předmět: |
Vertex (graph theory)
Group (mathematics) General Mathematics 010102 general mathematics Geometric Topology (math.GT) [MATH] Mathematics [math] Group Theory (math.GR) 01 natural sciences Mathematics::Geometric Topology [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] Action (physics) Separable space Combinatorics Low complexity Mathematics - Geometric Topology Mathematics::Group Theory 0103 physical sciences FOS: Mathematics Graph (abstract data type) 010307 mathematical physics [MATH]Mathematics [math] 0101 mathematics Mathematics - Group Theory [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR] Mathematics |
| DOI: | 10.48550/arxiv.1812.09715 |
| Popis: | Let $\Sigma_{g,p}$ be the genus--$g$ oriented surface with $p$ punctures, with either $g>0$ or $p>3$. We show that $MCG(\Sigma_{g,p})/DT$ is acylindrically hyperbolic where $DT$ is the normal subgroup of the mapping class group $MCG(\Sigma_{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma_{g,p}$ for suitable $K$. Moreover, we show that in low complexity $MCG(\Sigma_{g,p})/DT$ is in fact hyperbolic. In particular, for $3g-3+p\leq 2$, we show that the mapping class group $MCG(\Sigma_{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma_{g,p})$ is separable. The aforementioned results follow from general theorems about composite rotating families that come from a collection of subgroups of vertex stabilisers for the action of a group $G$ on a hyperbolic graph $X$. We give conditions ensuring that the graph $X/N$ is again hyperbolic and various properties of the action of $G$ on $X$ persist for the action of $G/N$ on $X/N$. Comment: 26 pages, 1 figure |
| Databáze: | OpenAIRE |
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