On the Subgroups of Right Angled Artin Groups and Mapping Class Groups

There exist right angled Artin groups $A$ such that the isomorphism problem
for finitely presented subgroups of $A$ is unsolvable, and for certain finitely
presented subgroups the conjugacy and membership problems are unsolvable. It
follows that if $S$ is a surface of finite type and the genus of $S$ is
sufficiently large, then the corresponding decision problems for the mapping
class group $Mod(S)$ are unsolvable. Every virtually special group embeds in
the mapping class group of infinitely many closed surfaces. Examples are given
of finitely presented subgroups of mapping class groups that have infinitely
many conjugacy classes of torsion elements.