Big mapping class groups with hyperbolic actions: classification and applications
Abstract
We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that Map() admits a continuous nonelementary action on a hyperbolic space if and only if contains a finite-type subsurface which intersects all its homeomorphic translates. When contains such a nondisplaceable subsurface K of finite type, the hyperbolic space we build is constructed from the curve graphs of K and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of Map() contains an embedded 1; second, using work of Dahmani, Guirardel and Osin, we deduce that Map() contains nontrivial normal free subgroups (while it does not if has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal.