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 Teichmüller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus surfaces (for any at least 2) in the moduli space of genus surfaces (for any at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.
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Matt T. Clay, Christopher J. Leininger, Johanna Mangahas, The geometry of right-angled Artin subgroups of mapping class groups. Groups Geom. Dyn. 6 (2012), no. 2, pp. 249–278DOI 10.4171/GGD/157