Abstract
AbstractGiven a lattice Veech group in the mapping class group of a closed surface , this paper investigates the geometry of , the associated ‐extension group. We prove that 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 on a hyperbolic space, retaining most of the geometry of . This action is a key ingredient in the sequel where we show that is hierarchically hyperbolic and quasi‐isometrically rigid.
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