Abstract
In this paper, we prove a combination theorem for indicable subgroups of
infinite-type (or big) mapping class groups. Importantly, all subgroups
produced by the combination theorem, as well as those coming 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 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 $\Z$, 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.