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 $\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 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 ${\mathrm {Map}}(\Sigma )$ 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.

Graphs of curves and arcs quasi-isometric to big mapping class groups

“…Another condition which we show to be equivalent to translatability is the following, due to Horbez, Qing, and Rafi [HQR20].…”

“…A non-connected surface is nondisplaceable if, for every f ∈ MCG(Σ) and every connected component S i of S, there is a connected component S j of S such that f (S i ) ∩ S j = ∅. Definition 5.2 (Definition 4.4 of [HQR20]). An avenue surface is a connected, orientable surface Σ which does not contain any nondisplaceable finite-type subsurfaces, whose end space is tame, and whose mapping class group MCG(Σ) admits a coarsely bounded generating set but is not itself coarsely bounded.…”

“…3 =⇒ 2: Lemma 4.5 of [HQR20] says that an avenue surface has zero or infinite genus and exactly two maximal ends, while Lemma 4.6 of [HQR20] says that every nonmaximal end of an avenue surface precedes both maximal ends under the standard ordering. It follows by Lemma 5.9 that Σ is translatable.…”

“…3. Σ has no nondisplaceable finite-type surfaces, making it an avenue surface in the sense of Horbez, Qing, and Rafi [HQR20].…”

“…We do not attempt in this paper to study the geometry of the translatable curve graph, although we hope this will be a fruitful avenue for further research. One property is however immediate: by the results of Horbez, Qing, and Rafi [HQR20] the translatable curve graph-and thus the mapping class group of a translatable surface-cannot be non-elementary δ-hyperbolic.…”