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Bowditch representations in Gromov-hyperbolic spaces : characterizations, dynamics of $\mathrm{Out}(\mathbb{F}_2)$ and recognition

Suzanne Schlich

TL;DR

This work generalizes Bowditch’s $BQ$-conditions from trace data to representations of $\mathbb{F}_2$ into isometry groups of Gromov-hyperbolic spaces by using the stable norm $l_S$ and a universal constant $K_\delta=433\delta$. It establishes a suite of equivalent characterizations for Bowditch representations, including linear lower bounds on $l_S(\rho(\gamma))$ in terms of word length and finiteness properties, and shows these representations form an open domain for Out($\mathbb{F}_2$) action with proper discontinuity. A key geometric ingredient is a large-scale inequality for products of hyperbolic isometries, replacing trace identities, together with Farey-tree methods to prove the core equivalences. The paper also links Bowditch representations to primitive-stable representations, and provides finite-certification algorithms for recognition, enabling practical verification within the geometric context. Overall, the results advance both the structural understanding and computational decidability of Bowditch phenomena in hyperbolic spaces.

Abstract

We study a generalization of the $BQ$-conditions, introduced by Bowditch and further developed by Tan-Wong-Zhang, for representations of the free group of rank two into isometry groups of Gromov-hyperbolic spaces. We show the existence of an explicit constant $K_δ$, depending only on the hyperbolicity constant $δ$ of the space, such that the hyperbolicity of the images of primitive elements together with the finiteness of the set of (conjugacy classes of) primitive elements whose images have lengths bounded by $K_δ$ imply a linear growth of the lengths with respect to the word length on primitive elements. We give several characterizations of Bowditch representations, and the framework developed allows us to prove that they form an open domain of discontinuity in the character variety. As a corollary, we also obtain a new characterization of primitive-stable representations, introduced by Minsky. Finally, we explain how our results can be used to obtain finite certificate for the recognition of Bowditch representations.

Bowditch representations in Gromov-hyperbolic spaces : characterizations, dynamics of $\mathrm{Out}(\mathbb{F}_2)$ and recognition

TL;DR

This work generalizes Bowditch’s -conditions from trace data to representations of into isometry groups of Gromov-hyperbolic spaces by using the stable norm and a universal constant . It establishes a suite of equivalent characterizations for Bowditch representations, including linear lower bounds on in terms of word length and finiteness properties, and shows these representations form an open domain for Out() action with proper discontinuity. A key geometric ingredient is a large-scale inequality for products of hyperbolic isometries, replacing trace identities, together with Farey-tree methods to prove the core equivalences. The paper also links Bowditch representations to primitive-stable representations, and provides finite-certification algorithms for recognition, enabling practical verification within the geometric context. Overall, the results advance both the structural understanding and computational decidability of Bowditch phenomena in hyperbolic spaces.

Abstract

We study a generalization of the -conditions, introduced by Bowditch and further developed by Tan-Wong-Zhang, for representations of the free group of rank two into isometry groups of Gromov-hyperbolic spaces. We show the existence of an explicit constant , depending only on the hyperbolicity constant of the space, such that the hyperbolicity of the images of primitive elements together with the finiteness of the set of (conjugacy classes of) primitive elements whose images have lengths bounded by imply a linear growth of the lengths with respect to the word length on primitive elements. We give several characterizations of Bowditch representations, and the framework developed allows us to prove that they form an open domain of discontinuity in the character variety. As a corollary, we also obtain a new characterization of primitive-stable representations, introduced by Minsky. Finally, we explain how our results can be used to obtain finite certificate for the recognition of Bowditch representations.

Paper Structure

This paper contains 32 sections, 69 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Characterizations of Bowditch representations in Gromov-hyperbolic spaces
  3. Dynamics of the outer automorphism group
  4. Recognition
  5. An ingredient of the proof
  6. Organization of the paper
  7. Preliminaries on hyperbolic spaces
  8. Shapes of triangles and quadrilaterals
  9. Gromov-boundary and shape of ideal-bigons and asymptotic geodesics
  10. Projections
  11. Hyperbolic isometries and stable norm
  12. Local-global Lemma
  13. Irreducibility of Bowditch representations
  14. Busemann functions
  15. Application of Busemann functions to representations
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\mathcal{X}$ be a geodesic, $\delta$-hyperbolic, visibility space. Let $K_\delta=433\delta$. Let $\rho : \mathbb{F}_2\to \mathrm{Isom}(\mathcal{X})$ be a representation. The following conditions are equivalent : When one (hence all) of these conditions is satisfied, we say that $\rho$ is a Bowditch representation.

Theorems & Definitions (147)

  • Definition 1.1: Bowditch bowditch_markoff_1998, Tan-Wong-Zhang tan_generalized_2008
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 1.7: Proposition \ref{['prop:ineq-l(AB)']} in section \ref{['sec:l(AB)']}
  • Lemma 2.1
  • proof
  • ...and 137 more