Affine diffeomorphism groups are undistorted

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.
Comment: 24 pages