A four point characterisation for coarse median spaces

TL;DR

This paper simplifies the definition of coarse median spaces using a 4-point condition, introduces a rank concept, and proves that rank 1 geodesic spaces are hyperbolic, enhancing understanding of their geometric structure.

Contribution

It provides a new 4-point characterization of coarse median spaces, defines rank in this context, and offers a direct proof that rank 1 spaces are hyperbolic, avoiding complex asymptotic cone arguments.

Findings

A 4-point condition characterizes coarse median spaces.

Rank 1 geodesic coarse median spaces are hyperbolic.

New definitions of intervals and their interactions with geodesics are introduced.

Abstract

Coarse median spaces simultaneously generalise the classes of hyperbolic spaces and median algebras, and arise naturally in the study of the mapping class groups and many other contexts. One issue with their definition as originally conceived by Bowditch is the need to establish median approximations for all finite subsets of the space, an approach which allowed the definition of rank (a proxy for dimension) in terms of the dimensions of the approximating spaces. Here we provide a simplification of the definition in terms of a $4$-point condition analogous to the $4$-point condition defining hyperbolicity. We show how to define rank in this context, and use this to give a direct proof that rank $1$ geodesic coarse median spaces are $\delta$-hyperbolic, bypassing Bowditch's use of asymptotic cones. A key ingredient of the proof is a new definition of intervals in coarse median spaces and an analysis of their interaction with geodesics.