2017/05/02 by Anthony Genevois, Genevois, Anthony
Mathematics · #20F65 #20F67 #43A15 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1705.00834
openalex publication_date 2017/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
From any two median spaces X,Y, we construct a new median space X \circledast Y, referred to as the diadem product of X and Y, and we show that this construction is compatible with wreath products in the following sense: given two finitely generated groups G,H and two (equivariant) coarse embeddings into median spaces X,Y, there exist a(n equivariant) coarse embedding G\wr H → X \circledast Y. As an application, we prove that α1(G \wr H) ≥ min(α1(G),α1(H))/2 for all finitely generated groups G,H, where α1(⋅) denotes the ℓ1-compression. As an other consequence, we recover several well-known theorems related to the Hilbertian geometry of wreath products from a unified point of view: the characterisation of wreath products satisfying Kazhdan's property (T) or the Haagerup property, as well as their discrete versions (FW) and (PW).