  Given a lattice Veech group in the mapping class group of a closed surface
$S$, this paper investigates the geometry of $\Gamma$, the associated
$\pi_1S$--extension group. We prove that $\Gamma$ is the fundamental group of a
bundle with a singular Euclidean-by-hyperbolic geometry. Our main result is
that collapsing "obvious" product regions of the universal cover produces an
action of $\Gamma$ on a hyperbolic space, retaining most of the geometry of
$\Gamma$. This action is a key ingredient in the sequel where we show that
$\Gamma$ is hierarchically hyperbolic and quasi-isometrically rigid.
