Right-angled Artin Groups and a Generalized Isomorphism Problem for Finitely Generated Subgroups of Mapping Class Groups

Consider the mapping class group Modg,p of a surface Σg,p of genus g with p punctures, and a finite collection {f1, . . . , fk} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N , the mapping classes {f 1 , . . . , f k } generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f1, . . . , fk} are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Modg,p, and recover the fact that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Modg,p.