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.Comment: 23 pages, v2: Minor revision following the referee's suggestions. The
final publication is available at link.springer.com via
https://doi.org/10.1007/s10711-015-0090-
