Kazhdan and Haagerup properties from the median viewpoint

Abstract

We prove the existence of a close connection between spaces with measured walls and median metric spaces.We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on L p -spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson and Steger.

Coarse median spaces and groups

“…One might conjecture that it applies to a much broader class of spaces that are in some sense non-positively curved, such as CAT(0) spaces. Much of this work is inspired by the results in [BehM1,BesBF,BehM2,BehBKM,BehDS,ChaDH]. It seems a natural general setting in which to view some of this work.…”

Median structures on asymptotic cones and homomorphisms into mapping class groups

“…Furthermore, there exists a vertex v in the geodesic t g shadowed by g for which v ∈ base(ρ), and hence satisfying Y ⊆ S \ v. Definition 2. 19. Given a constant K M (S), where M (S) is the constant from Lemma 2.18, and a pair of markings μ, ν, the subsurfaces Y ⊆ S, for which dist C(Y ) (μ, ν) > K, are called the K-large domains for the pair (μ, ν).…”

“…We denote by Y(F, G) the set of elements U = (U n ) ω in the ultrapower ΠU/ω such that, for any two points μ = lim ω (μ n ) ∈ F and ν = lim ω (ν n ) ∈ G, the subsurfaces U n are ω-a.s. K-large domains for the pair (μ n , ν n ), in the sense of Definition 2. 19.…”

“…The second statement is a consequence of the first. 19. Let (X i , dist i ) i∈I be a collection of metric spaces.…”

“…A point x is said to be between two other points a and b in a metric space (X, dist) if dist(a, x) + dist(x, b) = dist(a, b). See [19] for details.…”