  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.
