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.
Open books for Boothby-Wang bundles, fibered Dehn twists and the mean Euler characteristic
