Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups

TL;DR

This paper characterizes convex cocompact subgroups of mapping class groups via quasiconvexity in right-angled Artin groups, providing a new criterion and constructing subgroups with controlled orbit maps.

Contribution

It introduces a criterion linking convex cocompactness in mapping class groups to quasiconvexity in right-angled Artin groups, and constructs examples with small Lipschitz orbit maps.

Findings

Convex cocompact subgroups correspond to combinatorially quasiconvex subgroups in right-angled Artin groups.

Established conditions for right-angled Artin groups to be quasi-isometrically embedded in mapping class groups.

Constructed convex cocompact subgroups with orbit maps into the curve complex having small Lipschitz constants.

Abstract

We characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. That is, if the right-angled Artin group G in Mod(S) satisfies certain conditions that imply G is quasi-isometrically embedded in Mod(S), then a purely pseudo-Anosov subgroup H of G is convex cocompact in Mod(S) if and only if it is combinatorially quasiconvex in G. We use this criterion to construct convex cocompact subgroups of Mod(S) whose orbit maps into the curve complex have small Lipschitz constants.