A Note on the Norms of Transition Operators on Lamplighter Graphs and Groups

Let L X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the p-norms and spectral radii of P , if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the 2-spectral radius of symmetric random walks on such groups.