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Dehn filling Dehn twists. (English) Zbl 07316388

Summary: 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, in the sense of [13], that come from a collection of subgroups of vertex stabilizers 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 \(X/N\).

MSC:

20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
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