Affine diffeomorphism groups are undistorted

TL;DR

This paper proves that finitely generated affine diffeomorphism groups of half-translation surfaces are undistorted in the mapping class group and establishes a quasi-isometric embedding of the systole map into the curve complex.

Contribution

It demonstrates the undistorted nature of finitely generated affine diffeomorphism groups within the mapping class group and relates the systole map to the curve complex via quasi-isometry.

Findings

Finitely generated subgroups of affine diffeomorphism groups are undistorted.

The systole map is a quasi-isometric embedding into the curve complex.

Affine diffeomorphism groups stabilize Teichmüller discs.

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 Teichm\"uller disc under the action of the mapping class group on Teichm\"uller 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 Teichm\"uller disc is a quasi-isometric embedding into the curve complex.