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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.
We survey recent developments on mapping class groups of surfaces of infinite topological type.
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 th
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
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 endsp