Lamplighter groups, median spaces, and Hilbertian geometry

Abstract

From any two median spaces $X,Y$, we construct a new median space $X⊛Y$, 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 $G\wr H\to X⊛Y$. As an application, we prove that $${\alpha }_1(G\wr H)\ge \min ({\alpha }_1(G),{\alpha }_1(H))/2\text{ for all finitely generated groups G,H,}$$ where ${\alpha }_1(\cdot )$ denotes the ${\ell }^1$-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).

arXiv comment25 pages. Comments are welcome