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Register a new userBig mapping class groups: an overview
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We survey recent developments on mapping class groups of surfaces of infinite topological type.
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We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.
We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.
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.
In this paper, we prove a combination theorem for indicable subgroups of infinite-type (or big) mapping class groups. Importantly, all subgroups from the combination theorem, as well as those from the other results of the paper, can be constructed so that they do not lie in the closure of the compactly supported mapping class group and do not lie in the isometry group for any hyperbolic metric on the relevant infinite-type surface. Along the way, we prove an embedding theorem for indicable subgroups of mapping class groups, a corollary of which gives embeddings of pure big mapping class groups into other big mapping class groups that are not induced by embeddings of the underlying surfaces. We also give new constructions of free groups, wreath products with $mathbb Z$, and Baumslag-Solitar groups in big mapping class groups that can be used as an input for the combination theorem. One application of our combination theorem is a new construction of right-angled Artin groups in big mapping class groups.
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