  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
$\Mod_{g,p}$.
