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Manhattan geodesics and the boundary of the space of metric structures on hyperbolic groups

Stephen Cantrell, Eduardo Reyes

TL;DR

This paper develops a comprehensive coarse-geometric framework for metric structures on non-elementary hyperbolic groups. It introduces the Manhattan curve and a geodesic bicombing on the space of metric structures, proving that the optimal quasi-isometry constants between left-invariant, hyperbolic pseudometrics are given by dilations. It constructs a boundary, the Manhattan boundary, unifying metric structures from actions on CAT(0) cube complexes, real trees, and coned-off Cayley graphs, and embeds geodesic currents into this boundary, with extensions to stable translation lengths. The work further provides explicit Manhattan geodesics, proves continuity of stable translation length to currents in many cases, and presents a negative resolution of Bonahon’s conjecture by producing non-small actions with continuous translation-length extensions. Together these results connect Teichmüller-type boundaries, Outer space analogues, and current theory within a single, robust boundary framework for hyperbolic groups, with notable implications for Anosov representations and growth-rate analyses.

Abstract

For any non-elementary hyperbolic group $Γ$, we find an outer automorphism invariant geodesic bicombing for the space of metric structures on $Γ$ equipped with a symmetrized version of the Thurston metric on Techimüller space. We construct and study a boundary for this space and show that it contains many well-known pseudo metrics including those coming from actions on $\text{CAT}(0)$ cube complexes, real trees and coned-off Cayley graphs. As corollaries we deduce length spectrum rigidity results, regularity results for Manhattan curves, optimal growth rate results for Anosov representations and results regarding continuous extensions of translation distance functions to the space of geodesic currents. Using our results for geodesic currents we settle a conjecture of Bonahon in the negative.

Manhattan geodesics and the boundary of the space of metric structures on hyperbolic groups

TL;DR

This paper develops a comprehensive coarse-geometric framework for metric structures on non-elementary hyperbolic groups. It introduces the Manhattan curve and a geodesic bicombing on the space of metric structures, proving that the optimal quasi-isometry constants between left-invariant, hyperbolic pseudometrics are given by dilations. It constructs a boundary, the Manhattan boundary, unifying metric structures from actions on CAT(0) cube complexes, real trees, and coned-off Cayley graphs, and embeds geodesic currents into this boundary, with extensions to stable translation lengths. The work further provides explicit Manhattan geodesics, proves continuity of stable translation length to currents in many cases, and presents a negative resolution of Bonahon’s conjecture by producing non-small actions with continuous translation-length extensions. Together these results connect Teichmüller-type boundaries, Outer space analogues, and current theory within a single, robust boundary framework for hyperbolic groups, with notable implications for Anosov representations and growth-rate analyses.

Abstract

For any non-elementary hyperbolic group , we find an outer automorphism invariant geodesic bicombing for the space of metric structures on equipped with a symmetrized version of the Thurston metric on Techimüller space. We construct and study a boundary for this space and show that it contains many well-known pseudo metrics including those coming from actions on cube complexes, real trees and coned-off Cayley graphs. As corollaries we deduce length spectrum rigidity results, regularity results for Manhattan curves, optimal growth rate results for Anosov representations and results regarding continuous extensions of translation distance functions to the space of geodesic currents. Using our results for geodesic currents we settle a conjecture of Bonahon in the negative.
Paper Structure (25 sections, 54 theorems, 123 equations)

This paper contains 25 sections, 54 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Geodesics in the space of metric structures
  3. Boundary metric structures and the Manhattan boundary
  4. Examples of boundary metric structures
  5. Geodesic currents and a conjecture of Bonahon
  6. Preliminaries
  7. Gromov hyperbolic spaces and groups
  8. Hyperbolic groups
  9. The Manhattan curve
  10. Geodesic currents
  11. Optimal quasi-isometry constants
  12. Hyperbolic distance-like functions
  13. Proof of the main result
  14. Anosov representations
  15. Manhattan Geodesics
  16. ...and 10 more sections

Key Result

Theorem 1.1

For any $d,d_*\in \mathcal{D}_\Gamma$ there exists some $C\geq 0$ such that for all $x,y\in \Gamma$.

Theorems & Definitions (109)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 2.1
  • Proposition 2.2
  • ...and 99 more