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Gromov-Hausdorff convergence of maximal Gromov hyperbolic spaces and their boundaries

Kingshook Biswas, Arkajit Pal Choudhury

TL;DR

The paper studies continuity between two fundamental constructions in Gromov-hyperbolic geometry: boundary data and hyperbolic fillings. It proves that almost-isometric convergence of antipodal, compact quasi-metric spaces yields Gromov-Hausdorff convergence of the associated maximal Gromov hyperbolic spaces ${ m M}(Z)$, and that, under an equicontinuity hypothesis on boundaries, GH convergence of spaces implies AI convergence of their boundaries with visual metrics. The results unify the boundary–filling duality, show CAT$(-1)$ spaces inherit boundary convergence under GH limits, and deduce that finite-boundary maximal spaces sit nicely as AI/GH limits of polyhedral models. These continuity properties imply broader rigidity and stability results for Moebius maps, injective hulls, and CAT$(-1)$ fillable spaces, with concrete applications to polyhedral limits and boundary dynamics. Overall, the work provides a precise, technical bridge between convergence in the ambient hyperbolic spaces and convergence of their boundary structures via the Moebius and filling constructions.

Abstract

The relation between negatively curved spaces and their boundaries is important for various rigidity problems. In \cite{biswas2024quasi}, the class of Gromov hyperbolic spaces called maximal Gromov hyperbolic spaces was introduced, and the boundary functor $X \mapsto \partial X$ was shown to give an equivalence of categories between maximal Gromov hyperbolic spaces (with morphisms being isometries) and a class of compact quasi-metric spaces called quasi-metric antipodal spaces (with morphisms being Moebius homeomorphisms). The proof of this equivalence involved the construction of a filling functor $Z \mapsto {\mathcal M}(Z)$, associating to any quasi-metric antipodal space $Z$ a maximal Gromov hyperbolic space ${\mathcal M}(Z)$. We study the ``continuity" properties of the boundary and filling functors. We show that convergence of a sequence of quasi-metric antipodal spaces (in a certain sense called ``almost-isometric convergence") implies convergence (in the Gromov-Hausdorff sense) of the associated maximal Gromov hyperbolic spaces. Conversely, we show that convergence of maximal Gromov hyperbolic spaces together with a natural hypothesis of ``equicontinuity" on the boundaries implies convergence of boundaries. We use this to show that Gromov-Hausdorff convergence of a sequence of proper, geodesically complete CAT(-1) spaces implies Gromov-Hausdorff convergence of their boundaries equipped with visual metrics. We also show that convergence of maximal Gromov hyperbolic spaces to a maximal Gromov hyperbolic space with finite boundary implies convergence of boundaries.

Gromov-Hausdorff convergence of maximal Gromov hyperbolic spaces and their boundaries

TL;DR

The paper studies continuity between two fundamental constructions in Gromov-hyperbolic geometry: boundary data and hyperbolic fillings. It proves that almost-isometric convergence of antipodal, compact quasi-metric spaces yields Gromov-Hausdorff convergence of the associated maximal Gromov hyperbolic spaces , and that, under an equicontinuity hypothesis on boundaries, GH convergence of spaces implies AI convergence of their boundaries with visual metrics. The results unify the boundary–filling duality, show CAT spaces inherit boundary convergence under GH limits, and deduce that finite-boundary maximal spaces sit nicely as AI/GH limits of polyhedral models. These continuity properties imply broader rigidity and stability results for Moebius maps, injective hulls, and CAT fillable spaces, with concrete applications to polyhedral limits and boundary dynamics. Overall, the work provides a precise, technical bridge between convergence in the ambient hyperbolic spaces and convergence of their boundary structures via the Moebius and filling constructions.

Abstract

The relation between negatively curved spaces and their boundaries is important for various rigidity problems. In \cite{biswas2024quasi}, the class of Gromov hyperbolic spaces called maximal Gromov hyperbolic spaces was introduced, and the boundary functor was shown to give an equivalence of categories between maximal Gromov hyperbolic spaces (with morphisms being isometries) and a class of compact quasi-metric spaces called quasi-metric antipodal spaces (with morphisms being Moebius homeomorphisms). The proof of this equivalence involved the construction of a filling functor , associating to any quasi-metric antipodal space a maximal Gromov hyperbolic space . We study the ``continuity" properties of the boundary and filling functors. We show that convergence of a sequence of quasi-metric antipodal spaces (in a certain sense called ``almost-isometric convergence") implies convergence (in the Gromov-Hausdorff sense) of the associated maximal Gromov hyperbolic spaces. Conversely, we show that convergence of maximal Gromov hyperbolic spaces together with a natural hypothesis of ``equicontinuity" on the boundaries implies convergence of boundaries. We use this to show that Gromov-Hausdorff convergence of a sequence of proper, geodesically complete CAT(-1) spaces implies Gromov-Hausdorff convergence of their boundaries equipped with visual metrics. We also show that convergence of maximal Gromov hyperbolic spaces to a maximal Gromov hyperbolic space with finite boundary implies convergence of boundaries.

Paper Structure

This paper contains 24 sections, 51 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Antipodal spaces and their associated Moebius spaces
  4. Gromov product and Moebius spaces
  5. Injective metric spaces, maximal Gromov product spaces and geodesic bi-combing
  6. Semi-metric spaces and antipodal spaces
  7. Discrepancy, Antipodal flow and the Antipodalization map ${\mathcal{P}}_\infty$
  8. Partitions of unity on compact semi-metric spaces and smoothing operators
  9. Equicontinuous families of semi-metric spaces
  10. Gromov-Hausdorff convergence
  11. Gromov-Hausdorff-like distance between compact semi-metric spaces
  12. Proof of \ref{['forward']}
  13. Proof of \ref{['backward']}
  14. Construction of the candidate for almost isometry $g_n$
  15. The constructed candidate $g_n$ has small distortion for all $n$ sufficiently large
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\{(Z_n,\rho_n)\}_{n\ge 1}$ be a sequence of antipodal spaces which converge to an antipodal space $(Z,\rho_0)$ in the sense of almost-isometric convergence. Then the Moebius spaces $({\mathcal{M}}(Z_n),\rho_n)$ converge to the Moebius space $({\mathcal{M}}(Z),\rho_0)$ in Gromov-Hausdorff sense.

Theorems & Definitions (106)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Definition 2.1.1: Separating function or Semi-metric
  • Lemma 2.1.2: Geometric Mean- Value Theorem, biswas2015moebius,biswas2024quasi Lemma 2.2
  • ...and 96 more