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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Benjamini and G. Kozma: A resistance bound via an isoperimetric inequality, Combinatorica 25 (2005), 645–650.
I. Benjamini, H. Finucane and R. Tessera: On the scaling limit of finite vertex transitive graphs with large diameter. Combinatorica 36 (2016), 1–41.
E. Breuillard, B. J. Green and T. C. Tao: The structure of approximate groups, Publ. Math. IHES. 116 (2012), 115–221.
E. Breuillard and M. C. H. Tointon: Nilprogressions and groups with moderate growth, Adv. Math. 289 (2016), 1008–1055.
P. K. Carolino: The Structure of Locally Compact Approximate Groups. PhD thesis: https://escholarship.org/uc/item/8388n9jk.
P. Diaconis and L. Saloff-Coste: Moderate growth and random walk on finite groups, Geom. Funct. Anal. 4 (1994), 1–36.
R. Diestel and I. Leader: A conjecture concerning a limit of non-Cayley graphs, J. Algebraic Combin. 14 (2001), 17–25.
A. Eskin, D. Fisher and K. Whyte: Quasi-isometries and rigidity of solvable groups, Pure Appl. Math. Q. 3 (2007), Special Issue: In honor of Grigory Margulis. Part 1, 927–947.
M. Gromov: Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53–73.
M. Hall: The theory of groups, Amer. Math. Soc./Chelsea, Providence, RI (1999).
J. Hermon and R. Pymar: The exclusion process mixes (almost) faster than independent particles, arXiv:1808.10846v2.
E. Hewitt and K. A. Ross: Abstract Harmonic Analysis I (2nd ed.), Springer-Verlag, Berlin (1979).
B. Kleiner: A new proof of Gromov’s theorem on groups of polynomial growth, Jour. of the AMS 23 (2010), 815–829.
D. A. Levin and Y. Peres with contributions by E. L. Wilmer: Markov Chains and Mixing Times (2nd ed.), American Mathematical Society, Providence, RI (2017).
V. Losert: On the structure of groups with polynomial growth, Math. Z. 195 (1987), 109–117.
N. Ozawa: A functional analysis proof of Gromov’s polynomial growth theorem, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 549–556.
Y. Shalom and T. C. Tao: A finitary version of Gromov’s polynomial growth theorem, Geom. Funct. Anal. 20 (2010), 1502–1547.
T. C. Tao: Product set estimates for non-commutative groups, Combinatorica 28 (2008), 547–594.
T. C. Tao: Inverse theorems for sets and measures of polynomial growth, Q. J. Math. 68 (2017), 13–57.
R. Tessera and M. C. H. Tointon: Properness of nilprogressions and the persistence of polynomial growth of given degree, Discrete Anal. 17 (2018).
R. Tessera and M. C. H. Tointon: Sharp relations between volume growth, isoperimetry and resistance in vertex-transitive graphs, arXiv:2001.01467.
R. Tessera and M. C. H. Tointon: Lie group approximations of balls with polynomial volume in vertex-transitive graphs, in preparation.
M. C. H. Tointon: Freiman’s theorem in an arbitrary nilpotent group, Proc. London Math. Soc. (3) 109 (2014), 318–352.
M. C. H. Tointon: Introduction to approximate groups, London Mathematical Society Student Texts 94, Cambridge University Press, Cambridge (2020).
V. I. Trofimov: Graphs with polynomial growth, Math. USSR-Sb. 51 (1985), 405–417.
W. Woess: Topological groups and infinite graphs, Discrete Math. 95 (1991), 373–384.
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-020-4295-6