Stochastic approximation of lamplighter metrics

Abstract

We observe that embeddings into random metrics can be fruitfully used to study the L 1 $L_1$ -embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L 1 $L_1$ -distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an n $n$ -point metric space embeds bi-Lipschitzly into L 1 $L_1$ with distortion O ( log n ) $O(\log n)$ . In particular, for every finite group G $G$ the lamplighter group H = Z 2 ≀ G $H=\mathbb {Z}_2\wr G$ bi-Lipschitzly embeds into L 1 $L_1$ with distortion O ( log log | H | ) $O(\log \log |H|)$ . In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L 1 $L_1$ of the lamplighter group over the d $d$ -dimensional infinite lattice Z d $\mathbb {Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d $d$ -dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.