Coarse medians and universal quasigeodesic cones

TL;DR

This paper demonstrates that universal quasigeodesic cones in coarse median spaces inherently possess a canonical coarse median structure, linking hierarchically hyperbolic spaces to coarse median frameworks.

Contribution

It establishes a canonical coarse median structure on universal quasigeodesic cones and applies this to hierarchically hyperbolic spaces, extending their median properties.

Findings

Universal quasigeodesic cones admit a canonical coarse median structure

Hierarchically hyperbolic spaces can be endowed with compatible coarse median structures

Provides a new perspective on the structure of hyperbolic and median spaces

Abstract

We show that any universal quasigeodesic cone of uniformly coarse median spaces admits a canonical coarse median structure. As an application, we recover a result of Bowditch which states that any hierarchically hyperbolic space admits a coarse median structure compatible with the projections to the hyperbolic factor spaces.