Extensions of Veech groups I: A hyperbolic action

Given a lattice Veech group in the mapping class group of a closed surface S$S$, this paper investigates the geometry of Γ$\Gamma$, the associated π1S$\pi _1S$‐extension group. We prove that Γ$\Gamma$ 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 Γ$\Gamma$ on a hyperbolic space, retaining most of the geometry of Γ$\Gamma$. This action is a key ingredient in the sequel where we show that Γ$\Gamma$ is hierarchically hyperbolic and quasi‐isometrically rigid.