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• W3025345955 abstract "Let $Sigma$ be a connected, orientable surface of infinite type. Assume that $Sigma$ contains a nondisplaceable subsurface $K$ of finite type, i.e. $K$ intersects each of its homeomorphic translates. Then the mapping class group $Map(Sigma)$ admits a continuous nonelementary isometric action on a hyperbolic space, constructed from the curve graphs of $K$ and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. This has several applications: first, the second bounded cohomology of $Map(Sigma)$ contains an embedded $ell^1$; second, using work of Dahmani, Guirardel and Osin, we deduce that $Map(Sigma)$ 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. Conversely, under the assumptions that $Map(Sigma)$ is generated by a coarsely bounded set together with a tameness condition on the endspace of $Sigma$, we show that $Map(Sigma)$ does not act continuously nonelementarily by isometries on a hyperbolic space if $Sigma$ does not contain any nondisplaceable subsurface of finite type." @default.
• W3025345955 created "2020-05-21" @default.
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• W3025345955 date "2020-05-01" @default.
• W3025345955 modified "2023-09-27" @default.
• W3025345955 title "Hyperbolic actions of big mapping class groups" @default.
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