# A finitary structure theorem for vertex-transitive graphs of polynomial growth

## TL;DR

This paper establishes a quantitative structure theorem for vertex-transitive graphs of polynomial growth, showing they have quotients with virtually nilpotent or abelian actions, with implications for random walk behavior and growth patterns.

## Contribution

It provides a finitary version of Trofimov's theorem, extending structural results to broader classes of vertex-transitive graphs and analyzing their growth and random walk properties.

## Key findings

- Graphs of large diameter have quotients with small fibers and virtually abelian automorphism groups.
- Vertex-transitive graphs of polynomial growth exhibit predictable growth patterns across scales.
- Random walks on such graphs have quadratic mixing and relaxation times in relation to diameter.

## Abstract

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