Affine diffeomorphism groups are undistorted

Abstract

The affine diffeomorphism group $\mathrm{Aff}(S,q)$ of a half-translation surface $(S,q)$ comprise the self-diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space. We prove that any finitely generated subgroup of $\mathrm{Aff}(S,q)$ is undistorted in the mapping class group. We also show that the systole map restricted to the associated electrified Nielsen core in the Teichmüller disc is a quasi-isometric embedding into the curve complex.

24 pages

References in corpus (1)

• Boundaries and automorphisms of hierarchically hyperbolic spaces

Cited by in corpus (2)

• Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity
• Free products of abelian groups in mapping class groups