A Finitary Structure Theorem for Vertex-Transitive Graphs of Polynomial Growth

We prove a quantitative, finitary version of Trofimov’s result that connected, locally finite vertex-transitive graph Γ polynomial growth admits quotient with fibres on which the action Aut(Γ) is virtually nilpotent vertex stabilisers. also present some applications. show finite, connected large diameter small abelian stabilisers bounded size. has moderate in sense Diaconis and Saloff-Coste, known to imply mixing relaxation times lazy random walk are quadratic diameter. These results extend Breuillard second author for Cayley graphs Finally, given exhibiting at single, sufficiently scale, we describe its subsequent scales, extending Tao an earlier our own graphs. In forthcoming work will give further