Homological stability for subgroups of surface braid groups

Abstract

In this paper we prove homological stability for certain subgroups of surface braid groups. Alternatively, this is equivalent to proving homological stability for configurations of subsets of exactly $\xi$ points in a surface as we increase the number of subsets. For open surfaces, we prove the result integrally using a variation of the arc complex which we dub the "fern complex". We use a technique of Randal-Williams to extend the result rationally for closed surfaces.