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Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity

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

Abstract

We show that for any lattice Veech group in the mapping class group $\mathrm{Mod}(S)$ of a closed surface $S$, the associated $π_1 S$--extension group is a hierarchically hyperbolic group. As a consequence, we prove that any such extension group is quasi-isometrically rigid.

Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity

Abstract

We show that for any lattice Veech group in the mapping class group of a closed surface , the associated --extension group is a hierarchically hyperbolic group. As a consequence, we prove that any such extension group is quasi-isometrically rigid.

Paper Structure

This paper contains 32 sections, 63 theorems, 88 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Hierarchical hyperbolicity
  3. The HHG structure on $\Gamma$
  4. Quasi-isometric rigidity
  5. Motivation and Geometric Finiteness
  6. Outline and proofs
  7. Setup: The groups and spaces
  8. Flat metrics and Veech groups
  9. The bundles $E$ and $\bar{E}$.
  10. The hyperbolic space $\hat{E}$
  11. Vertices, spines, and spine bundles
  12. Schematic of the space $\bar{E}$ and its important pieces.
  13. Some technical lemmas and coarse geometry
  14. Projections and vertex spaces
  15. Quasimorphism distances
  16. ...and 17 more sections

Key Result

Theorem 1.1

For any lattice Veech subgroup $G < \mathrm{Mod}(S)$, the associated $\pi_1S$--extension group $\Gamma$ of $G$ is a hierarchically hyperbolic group.

Figures (10)

  • Figure 1: A schematic of $\bar{E}$ and various key features of it.
  • Figure 2: The surface obtained by gluing sides of the "L-shaped" polygon in pairs by translation according to the numbering has a decomposition into two cylinders (shaded blue and green) in the horizontal direction, $\alpha$; $\tau_\alpha$ is a twist in the bottom cylinder and a square of a twist in top cylinder. The boundaries of these cylinders (drawn in bold) form spines for the complement of core curves (drawn as dotted lines).
  • Figure 3: The window (shown in red) for $x$ in the thickend spine ${\bf\Theta}^v_Y$
  • Figure 4: Examples of bridges (in red), and the proof of Lemma \ref{['lem:bridge']}.
  • Figure 5: The proof of Lemma \ref{['lem:crossing spines']}: The arcs shown in red are the segments $\delta_i$, which contain $\Pi^v_Y(x)$ for all $x\in \theta^w$ with $f_Y(x)$ in the subgraph $W_i$ of $\theta^w_Y$.
  • ...and 5 more figures

Theorems & Definitions (128)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9: Tang
  • Definition 1.10: Parabolic geometric finiteness
  • ...and 118 more