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.
We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most 2. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most 4. Our proof relies on understanding how surgeries affect the Guirardel core associated to sphere systems. We show that applying a surgery is equivalent to performing a Rips move on the Guirardel core. arXiv:1610.05832v3 [math.GR]
It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. In [4], Croke and Kleiner present a right-angled Artin group with more than one visual boundary. In this paper we present a right-angled Coxeter group with non-unique equivariant visual boundary. The main theorem is that if right-angled Coxeter groups act geometrically on a Croke-Kleiner spaces constructed in [4], then the local angles in those spaces all have to be π/2. We present a specific right-angled Coxeter group with non-unique equivariant visual boundary. However, we conjecture that the right angled Coxeter groups that can act geometrically on a given CAT(0) space are far from unique.This implies that the "right-angled" in the terminology "right-angled Coxeter group" turns out to be literal and is consistent with the "geometric" property of the group. Furthermore, given the result from [3] that if we fix the gluing angle of the Croke-Kleiner space at π/2 and change the side lengths of the tori, the resulting boundaries are not equivariantly homeomorphic to each other, we conclude the following: Corollary 1.2. There exists a right-angled Coxeter group that does not have unique equivariant visual boundary.
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