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

TL;DR

This paper explores the complexity of decision problems in right angled Artin groups and mapping class groups, showing that certain problems are unsolvable and providing examples of complex subgroup structures.

Contribution

It demonstrates the existence of subgroups with unsolvable decision problems and shows how virtually special groups embed into mapping class groups.

Findings

Existence of right angled Artin groups with unsolvable subgroup problems.

Mapping class groups have subgroups with infinitely many conjugacy classes of torsion elements.

Embedding of virtually special groups into mapping class groups.

Abstract

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.