Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups (1007.1118v4)

Abstract: Consider the mapping class group $\Mod_{g,p}$ of a surface $\Sigma_{g,p}$ of genus $g$ with $p$ punctures, and a finite collection ${f_1,...,f_k}$ 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 ${f_1^N,...,f_k^N}$ 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 ${f_1,...,f_k}$ 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 $\Mod_{g,p}$, and prove that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of $\Mod_{g,p}$.