Constructing reducibly geometrically finite subgroups of the mapping class group

In this article, we consider qualified notions of geometric finiteness in mapping class groups called parabolically geometrically finite (PGF) and reducibly geometrically finite (RGF). We examine several constructions of subgroups and determine when they produce a PGF or RGF subgroup. These results provide a variety of new examples of PGF and RGF subgroups. Firstly, we consider the right-angled Artin subgroups constructed by Koberda (2012) and Clay–Leininger–Mangahas (2012), which are generated by high powers of given elements of the mapping class group. We give conditions on the supports of these elements that imply the resulting right-angled Artin subgroup is RGF. Secondly, we prove combination theorems which provide conditions for when a collection of reducible subgroups, or sufficiently deep finite-index subgroups thereof, generate an RGF subgroup.