On uniqueness of coarse median structures

TL;DR

This work shows that a finite product of bushy hyperbolic spaces has a unique coarse median structure, and that this uniqueness is preserved under relative hyperbolicity. It develops a robust framework linking coarse median spaces, ultralimits, and hyperbolic geometry, and uses Bowditch-style arguments to transfer median uniqueness from pieces to assembled spaces. The results yield new instances of spaces with unique coarse median structures, notably low-complexity pants graphs, and clarify the difference between canonical product medians and non-unique constructions in certain products. Overall, the paper advances understanding of when coarse median structures are canonical and how product and relative hyperbolicity geometry enforce rigidity in coarse median operators.

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.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Corollary 3
• proof
• Proposition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4: Bow-quasiflats, Lemma 10.5
• Proposition 2.5