# Circumcenter extension maps for non-positively curved spaces

## TL;DR

The paper introduces a new map called circumcenter extension for non-positively curved spaces and analyzes its properties.

## Contribution

The novel contribution is the extension of cross ratio preserving homeomorphisms to circumcenter extensions with Hölder-continuity and rough isometry properties.

## Key findings

- Circumcenter extensions exist under visibility conditions on Hadamard manifolds.
- The map is Hölder-continuous on specific regions and a rough isometry under cocompact group actions.
- Improved quasi-isometry constants are provided for 2-dimensional CAT(-1) manifolds.

## Abstract

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We describe regions on which this map is Hölder-continuous. Furthermore, we show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve previously known quasi-isometry constants, provided by Biswas, in the case of 2-dimensional \documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {CAT(-1)}$$\end{document}CAT(-1) manifolds. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.