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Isoperimetric profiles and random walks on some groups defined by piecewise actions

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Abstract

We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple gluing of such. In one of our simplest constructions—the pocket-extension of a group G—this leads to the study of certain finitely generated subgroups of the full permutation group \({\mathbb {S}}(G\cup \{*\})\). Some sharp estimates are obtained while many challenging questions remain.

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References

  1. Amir, Gideon: On the joint behaviour of speed and entropy of random walks on groups. Groups Geom. Dyn. 11(2), 455–467 (2017)

    Article  MathSciNet  Google Scholar 

  2. Amir, Gideon, Angel, Omer, Matte Bon, Nicolás, Virág, Bálint: The Liouville property for groups acting on rooted trees. Ann. Inst. Henri Poincaré Probab. Stat. 52(4), 1763–1783 (2016)

    Article  MathSciNet  Google Scholar 

  3. Amir, Gideon, Angel, Omer, Virág, Bálint: Amenability of linear-activity automaton groups. J Eur Math Soc 15(3), 705–730 (2013)

    Article  MathSciNet  Google Scholar 

  4. Amir, G., Kozma, G.: Groups with minimal harmonic functions as small as you like (With an appendix by Nicolas Matte Bon). arXiv e-prints, page arXiv:1605.07593, (May 2016)

  5. Amir, G., Kozma, G.: Every exponential group supports a positive harmonic function . Proc. Am. Math. Soc. (2017). https://doi.org/10.1090/proc/15435

  6. Amir, Gideon, Virág, Bálint: Positive speed for high-degree automaton groups. Groups Geom. Dyn. 8(1), 23–38 (2014)

    Article  MathSciNet  Google Scholar 

  7. Bartholdi, Laurent, Erschler, Anna: Poisson-Furstenberg boundary and growth of groups. Probab. Theory Related Fields 168(1–2), 347–372 (2017)

    Article  MathSciNet  Google Scholar 

  8. Bartholdi, Laurent, Kaimanovich, Vadim A, Nekrashevych, Volodymyr V: On amenability of automata groups. Duke Math. J. 154(3), 575–598 (2010)

    Article  MathSciNet  Google Scholar 

  9. Bartholdi, Laurent, Virág, Bálint: Amenability via random walks. Duke Math. J. 130(1), 39–56 (2005)

    Article  MathSciNet  Google Scholar 

  10. Bellman, Richard: Limit theorems for non-commutative operations I. Duke Math. J. 21, 491–500 (1954)

    MathSciNet  MATH  Google Scholar 

  11. Brieussel, Jérémie: Amenability and non-uniform growth of some directed automorphism groups of a rooted tree. Math. Z. 263(2), 265–293 (2009)

    Article  MathSciNet  Google Scholar 

  12. Brieussel, Jérémie, Zheng, Tianyi: Speed of random walks, isoperimetry and compression of finitely generated groups. Ann. Math. (2) 193(1), 1–105 (2021)

    Article  MathSciNet  Google Scholar 

  13. Brunel, A.: editor. Jourées sur les marches Aléatoires, (Nancy 1979). Astérisque 74, Société Mathématique de France, (1980)

  14. Coulhon, Thierry: Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141(2), 510–539 (1996)

    Article  MathSciNet  Google Scholar 

  15. Diaconis, Persi, Shahshahani, Mehrdad: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981)

    Article  MathSciNet  Google Scholar 

  16. Erschler, Anna: On drift and entropy growth for random walks on groups. Ann. Probab. 31(3), 1193–1204 (2003)

    Article  MathSciNet  Google Scholar 

  17. Erschler, Anna: On isoperimetric profiles of finitely generated groups. Geom. Dedicata 100, 157–171 (2003)

    Article  MathSciNet  Google Scholar 

  18. Erschler, Anna: Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks. Probab. Theory Related Fields 136(4), 560–586 (2006)

    Article  MathSciNet  Google Scholar 

  19. Erschler, Anna: Poisson-Furstenberg boundary of random walks on wreath products and free metabelian groups. Comment. Math. Helv. 86(1), 113–143 (2011)

    Article  MathSciNet  Google Scholar 

  20. Erschler, Anna: Almost invariance of distributions for random walks on groups. Probab. Theory Related Fields 174(1–2), 445–476 (2019)

    Article  MathSciNet  Google Scholar 

  21. Erschler, Anna, Zheng, Tianyi: Growth of periodic Grigorchuk groups. Invent. Math. 219(3), 1069–1155 (2020)

    Article  MathSciNet  Google Scholar 

  22. Frisch, Joshua, Hartman, Yair, Tamuz, Omer, Vahidi Ferdowsi, Pooya: Choquet-Deny groups and the infinite conjugacy class property. Ann. of Math. (2) 190(1), 307–320 (2019)

    Article  MathSciNet  Google Scholar 

  23. Guivarc’h, Y., Keane, M., Roynette, B.: Marches aléatoires sur les groupes de Lie. Lecture Notes in Mathematics, vol. 624. Springer, Berlin, New York (1977)

    Book  Google Scholar 

  24. Helfgott, Harald A., Seress, Ákos, Zuk, Andrzej: Random generators of the symmetric group: diameter, mixing time and spectral gap. J. Algebra 421, 349–368 (2015)

    Article  MathSciNet  Google Scholar 

  25. Houghton, C.H.: The first cohomology of a group with permutation module coefficients. Arch. Math. (Basel), 31(3):254–258, (1978/79)

  26. Michał, K., Bálint, V.: Non-Liouville groups with return probability exponent at most 1/2. Electron Commun. Probab. 20(12), 12 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Lee, J .R., Peres, Y.: Harmonic maps on amenable groups and a diffusive lower bound for random walks. Ann. Probab. 41(5), 3392–3419 (2013)

    Article  MathSciNet  Google Scholar 

  28. Lee, S. R.: Geometry of Houghton’s Groups. arXiv e-prints, page arXiv:1212.0257, (Dec 2012)

  29. Naor, A., Peres, Y.: Embeddings of discrete groups and the speed of random walks. Int. Math. Res. Not. (2008). https://doi.org/10.1093/imrn/rnn076

  30. Pittet, C., Saloff-Coste, L.: On random walks on wreath products. Ann. Probab. 30(2), 948–977 (2002)

    Article  MathSciNet  Google Scholar 

  31. Pittet, Ch., Saloff-Coste, L.: On the stability of the behavior of random walks on groups. J. Geom. Anal. 10(4), 713–737 (2000)

    Article  MathSciNet  Google Scholar 

  32. Roichman, Y.: Upper bound on the characters of the symmetric groups. Invent. Math. 125(3), 451–485 (1996)

    Article  MathSciNet  Google Scholar 

  33. Roussel, S.: Phénomène de cutoff pour certaines marches aléatoires sur le groupe symétrique. Colloq. Math. 86(1), 111–135 (2000)

    Article  MathSciNet  Google Scholar 

  34. Saloff-Coste, Laurent: Probability on groups: random walks and invariant diffusions. Notices Amer. Math. Soc. 48(9), 968–977 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Saloff-Coste, L., Zheng, T.: Random walks on free solvable groups. Math. Z. 279(3–4), 811–848 (2015)

    Article  MathSciNet  Google Scholar 

  36. Saloff-Coste, L., Zheng, T.: Random walks and isoperimetric profiles under moment conditions. Ann. Probab. 44(6), 4133–4183 (2016)

    Article  MathSciNet  Google Scholar 

  37. Saloff-Coste, L., Zheng, T.: Isoperimetric profiles and random walks on some permutation wreath products. Rev. Mat. Iberoam. 34(2), 481–540 (2018)

    Article  MathSciNet  Google Scholar 

  38. Saloff-Coste, L., Zheng, T.: Random walks on finite symmetric groups defined by piecewise actions In preparation, (2021)

  39. Tessera, R.: Isoperimetric profile and random walks on locally compact solvable groups. Rev. Mat. Iberoam. 29(2), 715–737 (2013)

    Article  MathSciNet  Google Scholar 

  40. Varopoulos, N.T.: Random walks on soluble groups. Bull. Sci. Math. (2) 107(4), 337–344 (1983)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, lthaca, USA

    Laurent Saloff-Coste

  2. Department of Mathematics, University of California San Diego, San Diego, USA

    Tianyi Zheng

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  1. Laurent Saloff-Coste
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  2. Tianyi Zheng
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Correspondence to Laurent Saloff-Coste.

Additional information

This work is dedicated to the memory of Harry Kesten who, among his many outstanding contributions to mathematics, initiated the study of random walks on groups.

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L. Saloff-Coste: Partially supported by NSF grant DMS 1404435 and DMS 1707589.

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Saloff-Coste, L., Zheng, T. Isoperimetric profiles and random walks on some groups defined by piecewise actions. Probab. Theory Relat. Fields 181, 711–756 (2021). https://doi.org/10.1007/s00440-021-01067-z

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  • DOI: https://doi.org/10.1007/s00440-021-01067-z

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