BIG MAPPING CLASS GROUPS WITH HYPERBOLIC ACTIONS: CLASSIFICATION AND APPLICATIONS

Abstract

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