Median geometry for spaces with measured walls and for groups

TL;DR

This paper demonstrates that uniform lattices acting on products of real hyperbolic spaces can act on median spaces in a way that is both proper and cocompact, revealing deep connections between hyperbolic geometry and median geometry.

Contribution

It establishes the existence of proper cocompact actions of certain lattices on median spaces, and introduces a quasification of median geometry to analyze spaces close to median spaces.

Findings

Lattices in products of hyperbolic spaces act properly and cocompactly on median spaces.

Spaces at finite Hausdorff distance from median spaces are characterized via quasification.

Complex hyperbolic spaces are not at finite Hausdorff distance from median spaces.

Abstract

We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of compatibility possible with the median geometry. Our theorem is also relevant for potential Rips-type theorems for median spaces. The result follows from an analysis of a quasification of median geometry that provides a geometric characterization of spaces at finite Hausdorff distance from a median space. We explain how the case of complex hyperbolic metric spaces is different, and that such spaces cannot be at finite Hausdorff distance from a median space.