Stochastic approximation of lamplighter metrics
Abstract
We observe that embeddings into random metrics can be fruitfully used to study the L1-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 L1-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a n-point metric space embeds bi-Lipschitzly into L1 with distortion O( n). In particular, for every finite group G the lamplighter group H = Z2 G bi-Lipschitzly embeds into L1 with distortion O(|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 L1 of the lamplighter group over the d-dimensional infinite lattice Zd can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.