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BIG MAPPING CLASS GROUPS WITH HYPERBOLIC ACTIONS: CLASSIFICATION AND APPLICATIONS
Published online by Cambridge University Press: 31 May 2021
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.
Keywords
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2173 - 2204
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
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