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Uniformization of intrinsic Gromov hyperbolic spaces

Vasudevarao Allu, Alan P Jose

Abstract

The purpose of this paper is to provide a uniformization procedure for Gromov hyperbolic spaces, which need not be geodesic or proper. We prove that the conformal deformation of a Gromov hyperbolic space is a bounded uniform space. Further, we show that there is a natural quasi-isometry between the Gromov boundary and the metric boundary of the deformed space. Our main results are a generalization of the results of Bonk, Heninonen, and Koskela [Proposition 4.5, Proposition 4.13, Astérisque 270 (2001)].

Uniformization of intrinsic Gromov hyperbolic spaces

Abstract

The purpose of this paper is to provide a uniformization procedure for Gromov hyperbolic spaces, which need not be geodesic or proper. We prove that the conformal deformation of a Gromov hyperbolic space is a bounded uniform space. Further, we show that there is a natural quasi-isometry between the Gromov boundary and the metric boundary of the deformed space. Our main results are a generalization of the results of Bonk, Heninonen, and Koskela [Proposition 4.5, Proposition 4.13, Astérisque 270 (2001)].

Paper Structure

This paper contains 8 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Arcs in metric space
  5. Conformal deformations
  6. Uniform spaces
  7. Gromov hyperbolic spaces
  8. Uniformization

Key Result

Theorem 1.1

The conformal deformations $X_\epsilon = \left(X, d_\epsilon\right)$ of a proper, geodesic $\delta$-hyperbolic space $X$ are bounded $A(\delta)-$uniform spaces for $0<\epsilon\leq \epsilon_0(\delta)$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2: Gromov hyperbolic space
  • Definition 2.3: $h-$short triangles
  • Lemma 2.4
  • Definition 2.5: Gromov sequences
  • Definition 2.6: Roads
  • Theorem 2.7
  • ...and 8 more