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Extensions of Veech groups I: A hyperbolic action

Spencer Dowdall, Matthew G. Durham, Christopher J. Leininger, Alessandro Sisto

Abstract

Given a lattice Veech group in the mapping class group of a closed surface $S$, this paper investigates the geometry of $Γ$, the associated $π_1S$--extension group. We prove that $Γ$ 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 $Γ$ on a hyperbolic space, retaining most of the geometry of $Γ$. This action is a key ingredient in the sequel where we show that $Γ$ is hierarchically hyperbolic and quasi-isometrically rigid.

Extensions of Veech groups I: A hyperbolic action

Abstract

Given 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.

Paper Structure

This paper contains 39 sections, 35 theorems, 105 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline and proofs
  3. Notation and guides
  4. Definitions and background
  5. Basic metric geometry
  6. Graph approximation
  7. Hyperbolicity
  8. Surfaces and mapping class groups
  9. Teichmüller spaces
  10. Quadratic differentials
  11. Flat geodesics, foliations, and directions
  12. Teichmüller disks
  13. Veech groups
  14. Setup and Notation
  15. Horoballs
  16. ...and 24 more sections

Key Result

Theorem 1.1

Suppose $G<\mathrm{Mod}(S)$ is a lattice Veech group with extension group $\Gamma$ and let $\Upsilon_1,\ldots,\Upsilon_k < \Gamma$ be representatives of the conjugacy classes of vertex subgroups. Then $\Gamma$ admits an isometric action on a hyperbolic space $\hat{E}$, quasi-isometric to the Cayley

Figures (5)

  • Figure 1: A cartoon of key aspects of $E$ and $\bar{E}$ over $D$ and $\bar{D}$, respectively.
  • Figure 2: A general triangle $\Delta$ in $E_0$. The nondegenerate subtriangle $\Delta'\subset \Delta$ is shown in bold.
  • Figure 3: Possible configurations of geodesic segments from $x$ to a cone point $w$ in the interior of $[y,z]$ (shown as thickened lines). The portion in the interior of $\Delta(x,y,z)$ is a single saddle connection. In the left-most picture $\nu$ is just the point $x$.
  • Figure 4: A fan $\Delta(x,y,z) \subset E_0$, in which $[f(x),f(y)]$ and $[f(y),f(z)]$ are both a single saddle connection.
  • Figure 5: Decomposing a triangle into fans based at pivot vertices.

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2: DDLSII
  • Theorem 1.3
  • Theorem 1.4: DDLSII
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2: Guessing geodesics
  • proof
  • Remark 2.3
  • ...and 67 more