# Lamplighter groups, median spaces, and Hilbertian geometry

## TL;DR

This paper introduces the diadem product of median spaces, demonstrating its compatibility with wreath products, and applies this to analyze coarse embeddings, -compression, and properties like Kazhdan's property (T) and the Haagerup property in wreath products.

## Contribution

The paper constructs the diadem product of median spaces and shows its compatibility with wreath products, providing new tools for analyzing geometric and algebraic properties of groups.

## Key findings

- -compression of wreath products is at least half the minimum of the factors' -compressions.
- Unified approach to characterizing properties like Kazhdan's property (T) and the Haagerup property in wreath products.
- Construction of a new median space operation compatible with group wreath products.

## Abstract

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 $G\wr H \to X \circledast Y$. As an application, we prove that $$\alpha_1(G \wr H) \geq \min(\alpha_1(G),\alpha_1(H))/2 \text{ for all finitely generated groups $G,H$,}$$ 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).