Dehn filling Dehn twists

Abstract

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 ⩽ 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.

Right-angled Artin groups as normal subgroups of mapping class groups

“…In analogy with this result, the first two authors prove in a separate paper that under certain hypotheses, the quotient complex arising in Theorem 1.6 is hyperbolic and that the quotient group is acylindrically hyperbolic [CM]. The argument follows along the lines of the work of Dahmani–Hagen–Sisto [DHS21].…”

“…As previously, denotes the normal subgroup of generated by th powers of Dehn twists. Building on the aforementioned work of Dahmani, it was recently shown by Dahmani–Hagen–Sisto that for suitable the quotient group is acylindrically hyperbolic [DHS21] (in particular, the group has infinite index in ). One of the major steps in the proof is to show that the quotient of the curve complex by is hyperbolic.…”

Hyperbolic quotients of projection complexes

“…A key feature of a projection complex is that, in general, it is a quasi-tree, in other words, it is quasi-isometric to a tree [1,Theorem 3.16]. Projection complexes have found several useful applications lately by many authors [2,3,[8][9][10][11][12].…”

Mapping class groups are quasicubical

“…The result has been an effusion of new understanding in both settings. For mapping class groups, this has included: confirmation of Farb's quasiflat conjecture [BHS21], semihyperbolicity [DMS20,HHP20], decision problems for subgroups [Bri13,Kob12], and residual properties [DHS21,BHMS20]; and on the cubical side: versions of Ivanov's theorem [Iva97,Fio20], characterisations of Morse geodesics [ABD21,IMZ21], control on purely loxodromic subgroups [KK14,KMT17], and results on uniform exponential growth [ANS `19].…”