Dehn filling Dehn twists

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 Kth 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(Sg,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.