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. A finitely presented subgroup of a mapping class group can have an exponential Dehn function.

Keywords

mapping class groups, RAAGs, virtually special groups, decision problems

2010 Mathematics Subject Classification

20F28, 20F65, 53C24, 57S25

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Published 3 December 2013