The geometry of right angled Artin subgroups of mapping class groups

TL;DR

This paper establishes conditions under which right-angled Artin groups can be embedded into mapping class groups in a way that preserves geometric structure, with implications for the geometry of moduli spaces.

Contribution

It provides new criteria for quasi-isometric embeddings of right-angled Artin groups into mapping class groups and demonstrates their geometric consequences in Teichmüller space.

Findings

Finite sets of mapping classes can generate right-angled Artin groups embedded quasi-isometrically.

Orbit maps to Teichmüller space are quasi-isometric embeddings under these conditions.

Infinitely many high-genus surfaces have universal covers quasi-isometrically embedded in Teichmüller space.

Abstract

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmuller space.