Coarse median structures and homomorphisms from Kazhdan groups

TL;DR

This paper introduces the concept of coarse median structures on metric spaces and demonstrates that groups with such structures have finitely many conjugacy classes of homomorphisms from Kazhdan groups, extending known results in hyperbolic and mapping class groups.

Contribution

It formalizes coarse median structures as equivalence classes and proves finiteness of conjugacy classes of homomorphisms from Kazhdan groups for these structures.

Findings

Groups with a uniformly left-invariant coarse median structure have finitely many conjugacy classes of homomorphisms from Kazhdan groups.

Extends classical approximation results of hyperbolic spaces by trees to coarse median spaces.

Generalizes known theorems about hyperbolic groups with property (T) and mapping class groups.

Abstract

We study Bowditch's notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan's property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock-Drutu-Sapir on the mapping class groups of orientable surfaces. We discuss a metric approximation property of finite subsets in coarse median spaces extending the classical result on approximation of Gromov hyperbolic spaces by trees.