A Quantitative Guessing Geodesics Theorem

TL;DR

This paper provides a quantitative version of the Guessing Geodesics theorem, offering explicit estimates of hyperbolicity constants and applying it to a hyperbolic model of CAT(0) spaces.

Contribution

It introduces a quantitative approach to Guessing Geodesics, explicitly estimating hyperbolicity constants, which enhances understanding of hyperbolic spaces.

Findings

Derived explicit hyperbolicity constant estimates

Applied the results to curtain models of CAT(0) spaces

Enhanced the quantitative understanding of hyperbolic metrics

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.