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Simple-stable and Bowditch representations of the four-punctured sphere group in Gromov-hyperbolic spaces

Suzanne Schlich

TL;DR

This work analyzes representations of the four-punctured sphere group π1(S_{0,4}) into isometries of a δ-hyperbolic space X. It introduces and relates two open domains—simple-stable representations and Bowditch representations—proving they are identical for the group in this setting. The core method blends combinatorial geometry of simple curves on S_{0,4} (via slope data and Farey/continued-fraction structure) with geometric group theory (uniform quasi-geodesicity and tubular-neighborhood control) to show Bowditch representations satisfy simple-stability. The results extend known parallels from rank-two free groups to surface groups, providing an open domain of discontinuity for the mapping class group action and broadening the understanding of convex-cocompact-type phenomena in Isom(X).

Abstract

In this paper, we study representations from the four-punctured sphere group into isometry groups of Gromov-hyperbolic spaces. We prove that the set of simple-stable representations (in analogy with Minsky's notion of primitive-stability) and the set of Bowditch representations (inspired by Bowditch's work on the free group of rank two, generalized by Tan, Wong and Zhang) are equal. Along the way, we study the combinatorics of simple closed curves on the four-punctured sphere and prove a result which quantifies the redundancy of subwords of certain given lengths within simple words.

Simple-stable and Bowditch representations of the four-punctured sphere group in Gromov-hyperbolic spaces

TL;DR

This work analyzes representations of the four-punctured sphere group π1(S_{0,4}) into isometries of a δ-hyperbolic space X. It introduces and relates two open domains—simple-stable representations and Bowditch representations—proving they are identical for the group in this setting. The core method blends combinatorial geometry of simple curves on S_{0,4} (via slope data and Farey/continued-fraction structure) with geometric group theory (uniform quasi-geodesicity and tubular-neighborhood control) to show Bowditch representations satisfy simple-stability. The results extend known parallels from rank-two free groups to surface groups, providing an open domain of discontinuity for the mapping class group action and broadening the understanding of convex-cocompact-type phenomena in Isom(X).

Abstract

In this paper, we study representations from the four-punctured sphere group into isometry groups of Gromov-hyperbolic spaces. We prove that the set of simple-stable representations (in analogy with Minsky's notion of primitive-stability) and the set of Bowditch representations (inspired by Bowditch's work on the free group of rank two, generalized by Tan, Wong and Zhang) are equal. Along the way, we study the combinatorics of simple closed curves on the four-punctured sphere and prove a result which quantifies the redundancy of subwords of certain given lengths within simple words.

Paper Structure

This paper contains 33 sections, 45 theorems, 85 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Simple-stability
  3. Bowditch representations
  4. Redundancy of subwords of simple words in $S_{0,4}$ and strategy of the proof
  5. Redundancy of subwords of simple words
  6. Strategy of the proof of Theorem \ref{['sph-BQ=PS']}
  7. Combinatorics of simple-closed curves on the four-punctured sphere
  8. The four-punctured sphere
  9. Simple closed curves on the four-punctured sphere
  10. Constructing simple closed curves
  11. Study of the redundancy of subwords of simple words
  12. Some generalities about lattices in $\mathbb{R}^2$
  13. Rectangles adapted to a basis of a lattice and tiling of $\mathbb{R}^2$
  14. Segments at height $h$ in the rectangles $S(x,u,v)$ and $R(x,u,v)$
  15. A few reminders about Farey neighbours
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $X$ be a $\delta$-hyperbolic, geodesic and visibility space. The sets of Bowditch representations and of simple-stable representations of $\pi_1(\mathrm{S}_{0,4})$ in $\mathrm{Isom}(X)$ are equal.

Figures (21)

  • Figure 1: The four punctured-sphere $S_{0,4}$
  • Figure 2: The sphere as the quotient of the plane : $\mathrm{S}_{0,4} \simeq (\mathbb{R}^2 \setminus \mathbb{Z}^2) / (2\mathbb{Z}^2,\pm)$
  • Figure 3: Fundamental triangle and square for $\mathrm{S}_{0,4}$
  • Figure 4: The three generators of $\pi_1(\mathrm{S}_{0,4}) = \mathbb{F}_3 = \langle a,b,c \rangle$.
  • Figure 5: Reading the word $\gamma = cac^{-1}ba$ in the fundamental triangle
  • ...and 16 more figures

Theorems & Definitions (94)

  • Theorem 1.1
  • Definition 1.1
  • Definition 1.2: Simple-stability
  • Definition 1.3: Bowditch representations
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4: Special-lengths
  • Proposition 2.1
  • proof
  • Remark 2.2
  • ...and 84 more