A four point characterisation for coarse median spaces

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.