Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups

TL;DR

This paper demonstrates that large powers of certain mapping classes generate right-angled Artin groups and explores the isomorphism problem for subgroups of mapping class groups and RAAGs.

Contribution

It establishes conditions under which powers of mapping classes generate RAAGs and analyzes the isomorphism problem for these subgroups.

Findings

Large powers of specific mapping classes generate RAAGs.

The generated RAAGs can be determined from the topology of the mapping classes.

The isomorphism problem for RAAGs is solvable, unlike for subgroups of mapping class groups.

Abstract

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 $\Mod_{g,p}$.