Given a lattice Veech group in the mapping class group of a closed surface $S$, this paper investigates the geometry of $\mathrm{\Gamma }$, the associated ${\pi }_{1}S$-extension group. We prove that $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ on a hyperbolic space, retaining most of the geometry of $\mathrm{\Gamma }$. This action is a key ingredient in the sequel where we show that $\mathrm{\Gamma }$ is hierarchically hyperbolic and quasi-isometrically rigid.

中文翻译：

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Veech 群的扩展 I：双曲线作用

给定闭曲面映射类群中的格 Veech 群$\mathrm{小号}$, 本文研究了几何$\mathrm{\Gamma }$, 相关的${\pi }_{\mathrm{1个}}\mathrm{小号}$-扩展组。我们证明$\mathrm{\Gamma }$是具有奇异双曲欧几里德几何的丛的基本群。我们的主要结果是，折叠通用封面的“明显”产品区域会产生以下作用$\mathrm{\Gamma }$在双曲空间上，保留大部分的几何形状$\mathrm{\Gamma }$. 这个动作是我们展示的续集中的关键要素$\mathrm{\Gamma }$是分层双曲和准等距刚性的。