On the Subgroups of Right Angled Artin Groups and Mapping Class Groups
Martin R. Bridson

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 such that the isomorphism problem for finitely presented subgroups of is unsolvable, and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows that if is a surface of finite type and the genus of is sufficiently large, then the corresponding decision problems for the mapping class group 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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Algebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology
