be the genus–g oriented surface with p punctures, with either g > 0 or p > 3. We show that
is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group
powers of Dehn twists about curves in
for suitable K.
Moreover, we show that in low complexity
is in fact hyperbolic. In particular, for 3g − 3 + p ⩽ 2, we show that the mapping class group
is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some
space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of
The aforementioned results follow from general theorems about composite rotating families, in the sense of , 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.