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

Romain Tessera, Matthew C H Tointon

Research output: Contribution to journalArticle (Academic Journal)peer-review

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.
Original languageEnglish
Pages (from-to)263-298
Number of pages36
JournalCombinatorica
Volume41
Issue number2
DOIs
Publication statusPublished - 21 Apr 2021

Bibliographical note

Funding 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.

Publisher Copyright:
© 2021, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

Keywords

  • 05C75
  • 05C25
  • 05C81
  • 22D99

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