On uniqueness of coarse median structures

TL;DR

This paper proves the uniqueness of coarse median structures in certain hyperbolic space products and explores how this property behaves under relative hyperbolicity, contrasting with known cases like mapping class groups.

Contribution

It establishes the uniqueness of coarse median structures for products of bushy hyperbolic spaces and shows this property is preserved under relative hyperbolicity.

Findings

Product of bushy hyperbolic spaces has a unique coarse median structure

Uniqueness property is closed under relative hyperbolicity

Non-hyperbolic pants graphs can have unique coarse median structures

Abstract

We show that any product of bushy hyperbolic spaces has a unique coarse median structure, and that having a unique coarse median structure is a property closed under relative hyperbolicity. As a consequence, in contrast with the case of mapping class groups, there are non-hyperbolic pants graphs that have unique coarse median structures.