On the scaling limit of finite vertex transitive graphs with large diameter

Abstract

Let (X n ) be an unbounded sequence of finite, connected, vertex transitive graphs such that |X n |=O(diam(X n )q) for some q>0. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence (X n ) converges in the Gromov Hausdorff distance to some finite dimensional torus equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov’s theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If X n is only roughly transitive and |X n |=O diam(X n δ) for δ >1 sufficiently small, we prove, this time by elementary means, that (X n ) converges to a circle.

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References

  1. [1]

    H. Abels: Specker-kompaktifizierungen von lokal kompakten topologischen gruppen, Mathematische Zeitschrift 135 (1974), 325–361.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    V. N. Berestovskí: Homogeneous manifolds with an intrinsic metric. I, Sibirsk. Mat. Zh. 29 (1988), 17–29.

    MathSciNet  Article  Google Scholar 

  3. [3]

    V. N. Berestovskí: Homogeneous manifolds with an intrinsic metric. II, Sibirsk. Mat. Zh. 30 (1989), 14–28.

    MathSciNet  Google Scholar 

  4. [4]

    I. Benjamini and O. Schramm: Finite transitive graph embeddings into a hyperbolic metric space must stretch or squeeze, GAFA seminar, Springer LNM (2012), To appear.

    Google Scholar 

  5. [5]

    I. Benjamini and A. Yadin: Harmonic measure in the presence of a spectral gap, arXiv:1402.0156.

  6. [6]

    E. Breuillard: Geometry of groups of polynomial growth and shape of large balls, preprint.

  7. [7]

    E. Breuillard and B. Green: Approximate groups I: the torsion free nilpotent case, J. de l’Institut de Math. de Jussieu 2011.

    Google Scholar 

  8. [8]

    E. Breuillard, B. Green and T. Tao: The structure of approximate groups, Publ. Math. i.h.è.s. 116 (2012), 115–221.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    E. Breuillard and E. Le Donne: On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, to appear in Proc. Natl. Acad. Sci.

  10. [10]

    E. Breuillard and M. Tointon: Nilprogressions and groups with moderate growth, Adv. Math. 289 (2016), 1008–1055.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    M. Bridson and A. Haefliger: Metric Spaces of Non-Positive Curvature, Grundl. der Math. Wiss. 319, Springer Verlag, 1999.

    Google Scholar 

  12. [12]

    N. Bourbaki: Élèments de mathèmatiques, Groupes et algèbres de Lie, Chapitre 9, Springer, 1982.

    Google Scholar 

  13. [13]

    D. Burago, Y. Burago and S. Ivanov: A Course in Metric Geometry, American Mathematical Society (2001).

    Google Scholar 

  14. [14]

    C. Champetier: L’espace des groupes de type fini, Topology 39 (2000), 657–680.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    J. Cheeger and T. H. Colding: On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000), 37–74.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    W. W. Comfort and S. Negrepontis: The theory of ultrafilters, Berlin, New York, Springer-Verlag, 1974.

    Google Scholar 

  17. [17]

    M. DeVos and B. Mohar: Small separations in vertex-transitive graphs, Electronic Notes in Discrete Mathematics 24 (2006), 165–172.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    G. A. Freiman: Foundations of a structural theory of set addition, translated from the Russian, Translations of Mathematical Monographs, Vol 37, American Mathematical Society, Providence, R. I. (1973), vii+108.

  19. [19]

    K. Fukaya: Collapsing of Riemannian manifolds and eigenvalues of the laplace operator, Invent. Math. 87 (1987), 517–547.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    K. Fukaya: Hausdorff convergence of Riemannian manifolds and its applications, Recent topics in differential and analytic geometry, 143-238, Adv. Stud. Pure Math. 18-I, Academic Press, Boston, MA, (1990).

    Google Scholar 

  21. [21]

    K. Fukaya and T. Yamaguchi: The fundamental groups of almost nonnegatively curved manifolds, Annals of Mathematics 136 (1992), 253–333.

    MathSciNet  Article  MATH  Google Scholar 

  22. [22]

    T. Gelander: A metric version of the Jordan-Turing theorem, Enseign. Math. 59 (2013), 1–12.

    MathSciNet  Article  Google Scholar 

  23. [23]

    M. Gromov: Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53–73.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    M. Gromov: Carnot-Carathèodory spaces seen from within, in: Sub-Riemannian Geometry, Progress in Mathematics 144, edited by A. Bellaiche and J-J. Risler, 79–323, Birkauser (1996).

    Google Scholar 

  25. [25]

    M. Gromov: Metric structures for Riemannian and non-Riemannian spaces, volume 152 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes.

  26. [26]

    D. Kazhdan: On ε-representation, Israel J. Math. 43 (1982), 315–323.

    MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    K. Kuwae and T. Shioya: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom. 11 (2003), 599–673.

    MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    P. Lorimer: Vertex-transitive graphs: Symmetric graphs of prime valency, Journal of graph theory 8 (1984), 55–68.

    MathSciNet  Article  MATH  Google Scholar 

  29. [29]

    A. Lubotsky: Expander Graphs in Pure and Applied Mathematics, Bull. Amer. Math. Soc. 49 (2012), 113–162.

    MathSciNet  Article  Google Scholar 

  30. [30]

    B. Kleiner: A new proof of Gromov’s theorem on groups of polynomial growth, Jour. AMS 23 (2010), 815–829.

    MathSciNet  MATH  Google Scholar 

  31. [31]

    D. Montgomery and L. Zippin: Topological transformation groups, Interscience Publishers, New York-London, 1955.

    Google Scholar 

  32. [32]

    P. Pansu: Croissance des boules et des gèodèsiques fermèes dans les nilvariètès, Ergodic Theory Dyn. Syst. 3 (1983), 415–445.

    Article  MATH  Google Scholar 

  33. [33]

    F. Peter and H. Weyl: Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann. 97 (1927), 737–755.

    MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    C. E. Praeger, L. Pyber, P. Spiga and E. Szabó: The Weiss conjecture for locally primitive graphs with automorphism groups admitting composition factors of bounded rank, Proc. Amer. Math. Soc. 140 (2012), 2307–2318.

    MathSciNet  Article  MATH  Google Scholar 

  35. [35]

    I. Z. Ruzsa: Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379–388.

    MathSciNet  Article  MATH  Google Scholar 

  36. [36]

    T. Sanders: On the Bogolyubov-Ruzsa lemma, Anal. PDE 5 (2012), 627–655.

    MathSciNet  Article  MATH  Google Scholar 

  37. [37]

    Y. Shalom and T. Tao: A finitary version of Gromov’s polynomial growth theorem, Geom. Funct. Anal. 20 (2010), 1502–1547.

    MathSciNet  Article  MATH  Google Scholar 

  38. [38]

    A. M. Turing: Finite approximations to Lie groups, Annals of Math. 39 (1938), 105–111.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Itai Benjamini.

Additional information

This work was done while she was a student at the Weizmann Institute of Science, Rehovot, Israel.

Supported by ANR-09-BLAN-0059 and ANR-10-BLAN 0116.

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Benjamini, I., Finucane, H. & Tessera, R. On the scaling limit of finite vertex transitive graphs with large diameter. Combinatorica 37, 333–374 (2017). https://doi.org/10.1007/s00493-015-2975-4

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Mathematics Subject Classification (2000)

  • 20F65