A quantitative version of the Morse lemma and ideal boundary fixing quasiisometries

TL;DR

This paper provides a precise quantitative analysis of the Morse Lemma and its anti-version in hyperbolic spaces, offering optimal bounds and applications to quasi-isometries fixing the ideal boundary.

Contribution

It establishes the optimal upper bounds for the Morse and anti-Morse Lemmas and explores their applications in the context of ideal boundary fixing quasi-isometries.

Findings

Proves optimal bounds for Morse Lemma in hyperbolic spaces.

Establishes the anti-Morse Lemma with logarithmic neighborhood bounds.

Applies results to analyze point displacement under boundary-fixing quasi-isometries.

Abstract

The article is devoted to a proof of the optimal upper-bound for Morse Lemma, its "anti"-version and their applications. Roughly speaking, Morse Lemma states that in a hyperbolic metric space, a $\lambda$-quasi-geodesic $\gamma$ sits in a $\lambda^2$-neighborhood of every geodesic $\sigma$ with same endpoints. Anti-Morse Lemma states that $\sigma$ sits in a $\log\lambda$-neighborhood of $\gamma$. Applications include the displacement of points under quasi-isometries fixing the ideal boundary.