Extensions of Veech groups I: A hyperbolic action

Given a lattice Veech group in the mapping class of closed surface S $S$ , this paper investigates geometry Γ $\Gamma$ associated π 1 $\pi _1S$ -extension group. We prove that is fundamental bundle with singular Euclidean-by-hyperbolic geometry. Our main result collapsing “obvious” product regions universal cover produces an action on hyperbolic space, retaining most . This key ingredient sequel where we show hierarchically and quasi-isometrically rigid.