# On the bi-Lipschitz geometry of lamplighter graphs

## TL;DR

This paper systematically studies the bi-Lipschitz geometry of lamplighter graphs, showing embeddings into Hamming cubes and ℓ₁, and characterizing superreflexivity and clique presence through these graphs.

## Contribution

It provides new bi-Lipschitz embedding results for lamplighter graphs over trees and characterizes superreflexivity and clique presence via lamplighter graph properties.

## Key findings

- Lamplighter graphs over trees embed into Hamming cubes with distortion ≤6.
- Lamplighter graphs over countable trees embed into ℓ₁.
- Presence of a clique in a graph implies a Hamming cube in its lamplighter graph.

## Abstract

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most~$6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterization in terms of a sequence of graphs with uniformly bounded degree of the notion of trivial Bourgain-Milman-Wolfson type for arbitrary metric spaces, similar to Ostrovskii's characterization previously obtained in \cite{ostrovskii:11}.