## Abstract

We prove a quantitative, finitary version of Trofimov’s result that a connected, locally finite vertex-transitive graph *Γ* of polynomial growth admits a quotient with finite fibres on which the action of Aut(*Γ*) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph *Γ* of large diameter admits a quotient with fibres of small diameter on which the action of Aut(*Γ*) is virtually abelian with vertex stabilisers of bounded size. We also show that *Γ* 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 *Γ* 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 *Γ* 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.

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## Additional information

M. Tointon was supported by the Stokes Research Fellowship at Pembroke College, Cambridge; by grant FN 200021_163417/1 of the Swiss National Fund for scientific research; and by a travel grant from the Pembroke College Fellows’ Research Fund.

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Tessera, R., Tointon, M.C.H. A Finitary Structure Theorem for Vertex-Transitive Graphs of Polynomial Growth.
*Combinatorica* **41, **263–298 (2021). https://doi.org/10.1007/s00493-020-4295-6

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### Mathematics Subject Classification (2010)

- 05C75
- 05C25
- 05C81
- 22D99