Extensions of Veech groups I: A hyperbolic action
Extensions of Veech groups I: A hyperbolic action
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DOI：
10.1112/topo.12296
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发表时间：
2020-06
影响因子：
1.1
通讯作者：
S. Dowdall;Matthew G. Durham-;C. Leininger;A. Sisto
中科院分区：
文献类型：
--
作者：
S. Dowdall;Matthew G. Durham-;C. Leininger;A. Sisto
给定闭曲面S$S$的映射类群中的格Veech群，研究了相应的π 1 S $\pi_1S $-扩张群Γ$\Gamma$的几何.我们证明了Γ$\Gamma$是具有奇异Euclidean-by-bicycle几何的丛的基本群。我们的主要结果是，坍缩泛复盖的“明显”乘积区域产生了双曲空间上的Γ$\Gamma$的作用，保留了Γ$\Gamma$的大部分几何。这个作用量是我们证明Γ$\Gamma$是层次双曲和拟等距刚性的续篇中的关键成分。
Given a lattice Veech group in the mapping class group of a closed surface S$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.