Dehn filling Dehn twists

Abstract

Let Σg,p be the genus-g oriented surface with p punctures, with either g > 0 or p > 3. We show that M CG(Σg,p)/DT is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group M CG(Σg,p) generated by K th powers of Dehn twists about curves in Σg,p for suitable K.Moreover, we show that in low complexity M CG(Σg,p)/DT is in fact hyperbolic. In particular, for 3g − 3 + p ≤ 2, we show that the mapping class group M CG(Σg,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 M CG(Σg,p) is separable.The aforementioned results follow from general theorems about composite rotating families, in the sense of [Dah18], 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 . ContentsHagen was supported by EPSRC grant EPSRC EP/R042187/1.

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

“…Quotients by powers of Dehn twists. We first apply Theorem 1 to quotients of mapping class groups by large powers of Dehn twists, further developing the technology used in [Dah18,DHS18].…”

“…In [DHS18], it is proven that quotients of mapping class groups by large powers of Dehn twists are acylindrically hyperbolic, providing an analogue of the Dehn filling theorem. However, while acylindrical hyperbolicity has many interesting consequences, it only captures part of the geometry of mapping class groups.…”

“…A strategy to give a positive answer to Question 3 could involve adapting arguments [BHS17a, Section 5] and from [Bow15], and proving a combinatorial rigidity result. Using the lifting techniques from [DHS18] that we develop further in Subsection 8.7, it might be possible to prove such a result by reducing it to combinatorial rigidity of a suitable complex on which M CGpSq acts. We pointed to [Bow15] because we expect the structure of M CGpSq{DT K to more closely resemble that of the Weil-Petersson metric than that of the mapping class group itself, since, roughly speaking, we made the annular curve graphs bounded, hence irrelevant.…”

“…Hyperbolic quotients. In [DHS18], it is proven that M CGpΣ 0,5 q and M CGpΣ 1,2 q are fully residually non-elementary hyperbolic. We push this further, using Theorem 2 (together with the description of the resulting HHS structure) to show:…”

Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes

“…It was recently shown by Behrstock, Hagen and Sisto [BHS17b, BHS19a] that the Masur-Minsky machinery can be applied to a much vaster class of spaces, which they name hierarchically hyperbolic spaces. This has proved a fruitful approach to a number of problems [BHS17a,DHS17,DHS18], notably allowing the authors to obtain a particularly strong rigidity result for quasi-flats [BHS19b].…”