Extensions of Veech groups I: A hyperbolic action

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.