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.

我们解决了确定无限类型曲面的哪些映射类组允许双曲空间上的非初等连续作用的问题。

更准确地说，设 $\Sigma$ 是一个连接的、可定向的无限类型的曲面，它具有温和的端空间，其映射类组由粗限界子集生成。我们证明 ${\mathrm {Map}}(\Sigma )$ 允许在双曲空间上存在连续的非初等作用当且仅当 $\Sigma$ 包含一个有限类型的子曲面，该子曲面与其所有同胚平移相交。

当 $\Sigma$ 包含这样一个有限类型的不可位移地下K时，我们构建的双曲空间是从K的曲线图构造的，其同胚通过 Bestvina、Bromberg 和 Fujiwara 的构造进行平移。 我们的构造有几个应用：首先， ${\mathrm {Map}}(\Sigma )$ 的第二个有界上同调包含嵌入的 $\ell ^1$ ；其次，使用 Dahmani、Guirardel 和 Osin 的工作，我们推断 ${\mathrm {Map}} (\Sigma )$ 包含非平凡的正规自由子群（如果 $\Sigma $ 没有有限类型的不可位移次表面，则它不包含） , 有无数个商并且是 SQ-universal。