Stochastic approximation of lamplighter metrics

Abstract

We observe that embeddings into random metrics can be fruitfully used to study the 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 -distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an 𝑛-point metric space embeds bi-Lipschitzly into 𝐿 1 with distortion 𝑂(log 𝑛). In particular, for every finite group 𝐺 the lamplighter group 𝐻 = ℤ 2 ≀ 𝐺 bi-Lipschitzly embeds into 𝐿 1 with distortion 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 of the lamplighter group over the 𝑑-dimensional infinite lattice ℤ 𝑑 can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite 𝑑dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.