Circumcenter extension of Moebius maps to CAT(-1) spaces

TL;DR

This paper introduces a circumcenter-based extension of Moebius maps between boundaries of CAT(-1) spaces, establishing its properties as a quasi-isometry and its naturality under composition with isometries.

Contribution

It defines the circumcenter extension of Moebius maps for CAT(-1) spaces and proves its quasi-isometric properties and naturality, connecting boundary maps to space extensions.

Findings

The extension coincides with a known quasi-isometric extension.

It is locally 1/2-Holder continuous.

It is a quasi-isometry under curvature bounds.

Abstract

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ is shown to coincide with the $(1, \log 2)$-quasi-isometric extension constructed in [biswas3], and is locally $1/2$-Holder continuous. When $X,Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$ for some $b \geq 1$ then the extension $\hat{f} : X \to Y$ is a $(1, (1 - \frac{1}{b})\log 2)$-quasi-isometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries.