Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Quantitative Guessing Geodesics Theorem

Talia Shlomovich

Abstract

We present a quantitative version of Guessing Geodesics, which is a well-known theorem that provides a set of conditions to prove hyperbolicity of a given metric space. This version adds to the existing result by determining an explicit estimate of the hyperbolicity constant. As a sample application of this result, we estimate the hyperbolicity constant for a particular hyperbolic model of $\mathrm{CAT}(0)$ spaces known as the curtain model.

A Quantitative Guessing Geodesics Theorem

Abstract

We present a quantitative version of Guessing Geodesics, which is a well-known theorem that provides a set of conditions to prove hyperbolicity of a given metric space. This version adds to the existing result by determining an explicit estimate of the hyperbolicity constant. As a sample application of this result, we estimate the hyperbolicity constant for a particular hyperbolic model of spaces known as the curtain model.

Paper Structure

This paper contains 9 sections, 9 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Guessing Geodesics
  3. Quantitative Guessing Geodesics
  4. Estimation of $\kappa$
  5. Determining a Hyperbolicity Constant
  6. Application to the Curtain Model
  7. Background
  8. Determining Constants for $\mathrm{CAT}(0)$ Spaces
  9. Estimation of Hyperbolicity of Curtain Model

Key Result

Theorem A

Let $(X,d)$ be a geodesic metric space. Assume that for some constant $D > 0$ there are paths $\eta_{xy} = \eta(x, y) : [0, 1] \to X$ from $x$ to $y$, for each pair $x,y \in X$. Suppose also that the following conditions are satisfied: Then $(X,d)$ is $\delta$-hyperbolic for some $\delta = \delta(D) > 0$.

Figures (4)

  • Figure 1: The $D$-coarsely connected paths in (a) arise from a subdivision of the interval $[0,2^n]$ forming two paths, and (b) shows a subdivision forming $k$ paths.
  • Figure 2: The path $\lambda$ shown as a concatenation of three $(q_1,q_2)$--quasi-geodesic paths.
  • Figure 3: Dotted curves are used to denote quasi-geodesics. The blue curves denote the "fellow-travelling section", with the endpoints labelled accordingly.
  • Figure 4: Quadrilateral separated into two thin quadrilaterals and a "fellow-travelling section" between them, where the latter two sections are greyed out.

Theorems & Definitions (36)

  • Theorem A: Guessing Geodesics, UH
  • Theorem B: Quantitative Guessing Geodesics
  • Theorem C: Rough Guessing Geodesics
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Proposition 2.1.4
  • Remark
  • proof
  • proof : Proof of Claim 1.
  • ...and 26 more