The Liouville property and Hilbertian compression
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, p. 2435-2454
Lower bounds on the equivariant Hilbertian compression exponent α are obtained using random walks. More precisely, if the probability of return of the simple random walk is exp(-n γ ) in a Cayley graph then α(1-γ)/(1+γ). This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if |B n |e n ν then the speed exponent is 1/(2-ν).Under a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to α1-γ. Using a result from Naor & Peres on compression and the speed of random walks, this yields very promising bounds on speed and implies the Liouville property if γ<1/2.
Des bornes inférieures sur l’exposant de compression hilbertienne équivariante α sont données en utilisant les marches aléatoires. Plus précisément, si la probabilité de retour de la marche aléatoire est exp(-n γ ) pour un graphe de Cayley, alors α(1-γ)/(1+γ). Ceci motive l’étude de relations supplémentaires entre la probabilité de retour, la vitesse, l’entropie et la croissance du volume. Par exemple, if |B n |e n ν , alors l’exposant de vitesse est 1/(2-ν).Avec une hypothèse plus forte sur le comportement du noyau de la chaleur hors de la diagonale, la borne inférieure sur la compression α1-γ. Par un résultat de Naor et Peres sur la compression et la vitesse des marches aléatoires, ceci donne un estimé prometteur sur la vitesse et implique la propriété de Liouville si γ<1/2.
Received : 2014-08-05
Revised : 2016-01-22
Accepted : 2016-03-24
Published online : 2016-10-04
DOI : https://doi.org/10.5802/aif.3067
Classification:  20F69,  60J50,  60G50,  37A35,  43A65
Keywords: Hilbertian compression, random walks on groups, entropy, drift, growth of groups
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     author = {Gournay, Antoine},
     title = {The Liouville property and Hilbertian compression},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     pages = {2435-2454},
     doi = {10.5802/aif.3067},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2016__66_6_2435_0}
}
The Liouville property and Hilbertian compression. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2435-2454. doi : 10.5802/aif.3067. https://aif.centre-mersenne.org/item/AIF_2016__66_6_2435_0/

[1] Amir, Gideon On the joint behaviour of speed and entropy of random walks on groups (http://arxiv.org/abs/1509.00256 )

[2] Amir, Gideon; Virág, Bálint Speed exponents for random walks on groups (http://arxiv.org/abs/1203.6226 )

[3] Arzhantseva, G. N.; Guba, V. S.; Sapir, M. V. Metrics on diagram groups and uniform embeddings in a Hilbert space, Comment. Math. Helv., Tome 81 (2006) no. 4, pp. 911-929 | Article

[4] Austin, T. Amenable groups with very poor compression into Lebesgue spaces, Duke Math. J., Tome 159 (2011) no. 2, pp. 187-222 | Article

[5] Austin, T.; Naor, Assaf; Peres, Yuval The wreath product of with has Hilbert compression exponent 2/3, Proc. Amer. Math. Soc., Tome 137 (2009) no. 1, pp. 85-90 | Article

[6] Avez, A. Entropie des groupes de type fini, C. R. Acad. Sci. Paris Sér. A-B, Tome 275 (1972), p. A1363-A1366

[7] Avez, A. Théorème de Choquet-Deny pour les groupes à croissance non exponentielle, C. R. Acad. Sci. Paris Sér. A, Tome 279 (1974), pp. 25-28

[8] Bartholdi, Laurent; Erschler, Anna Imbeddings into groups of intermediate growth (http://arxiv.org/abs/1403.5584 )

[9] Bekka, B.; Chérix, A.; Valette, A. Proper affine isometric actions of amenable groups, Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 227 (1995), pp. 1-4

[10] Bendikov, A.; Pittet, C.; Sauer, R. Spectral distribution and L 2 -isoperimetric profile of Laplace operators on groups, Math. Ann., Tome 354 (2012), pp. 43-72 | Article

[11] Benyamini, Y.; Lindenstrauss, J. Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society, providence, American Mathematical Society Colloquium Publications, Tome 48 (2000)

[12] Brieussel, Jérémie; Zheng, Tianyi Speed of random walks, isoperimetry and compression of finitely generated groups (https://arxiv.org/abs/1510.08040 )

[13] Carette, Mathieu; Arnt, Sylvain; Thibault, Pillon; Valette, Alian The Haagerup property is not invariant under quasi-isometry (https://arxiv.org/abs/1403.5446 )

[14] Carne, T. K. A transmutation formula for Markov chains, Bull. Sci. Math., Tome 109 (1985) no. 4, pp. 399-405

[15] Coulhon, Thierry; Grigor’Yan, Alexander On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J., Tome 89 (1997) no. 1, pp. 133-199 | Article

[16] Coulhon, Thierry; Grigor’Yan, Alexander; Pittet, Christophe A geometric approach to on-diagonal heat kernels lower bounds on groups, Ann. Inst. Fourier, Tome 51 (2001) no. 6, pp. 1763-1827 | Article

[17] Coulhon, Thierry; Grigor’Yan, Alexander; Zucca, F. The discrete integral maximum principle and its applications, Tohoku Math. J., Tome 57 (2005) no. 4, pp. 447-621 | Article

[18] Dungey, N. Properties of random walks on discrete groups: Time regularity and off-diagonal estimates, Bull. Sci. math., Tome 132 (2008), pp. 359-381 | Article

[19] Erschler, Anna On drift and entropy growth for random walks on groups, Ann. Probab., Tome 31 (2003) no. 3, pp. 1193-1204 | Article

[20] Erschler, Anna Critical constants for recurrence of random walks on G-spaces, Ann. Inst. Fourier, Tome 55 (2005) no. 2, pp. 493-509 | Article

[21] Erschler, Anna; Karlsson, Anders Homomorphisms to constructed from random walks, Ann. Inst. Fourier, Tome 60 (2010) no. 6, pp. 2095-2113 | Article

[22] Guentner, E.; Kaminker, J. Exactness and uniform embeddability of discrete groups, J. London Math. Soc., Tome 70 (2004) no. 3, pp. 703-718 | Article

[23] Kaimanovich, V. Boundary behaviour of Thompson’s group (in preparation)

[24] Kesten, H. Full Banach mean values on countable groups, Math. Scand., Tome 7 (1959), pp. 146-156 | Article

[25] Kesten, H. Symmetric random walks on groups, Trans. Amer. Math. Soc., Tome 92 (1959), pp. 336-354 | Article

[26] Kotowski, Michał; Virág, Bálint Non-Liouville groups with return probability exponent at most 1/2 (http://arxiv.org/abs/1408.6895 )

[27] Lee, J.; Peres, Y. Harmonic maps on amenable groups and a diffusive lower bound for random walks, Ann. Probab., Tome 41 (2013) no. 5, pp. 3392-3419 | Article

[28] Naor, Assaf; Peres, Yuval Embeddings of discrete groups and the speed of random walks, Int. Math. Res. Not. (2008), 34 pages

[29] Naor, Assaf; Peres, Yuval L p -compression, traveling salesmen, and stable walks, Duke Math. J., Tome 157 (2011) no. 1, pp. 53-108 | Article

[30] Pittet, C.; Saloff-Coste, L. On the stability of the behavior of random walks on groups, J. Geom. Anal., Tome 10 (2000) no. 4, pp. 713-737 | Article

[31] Pittet, C.; Saloff-Coste, L. On random walks in wreath products, Ann. Probab., Tome 30 (2002) no. 2, pp. 948-977 | Article

[32] Revelle, D. Rate of escape of random walks on wreath products and related groups, Ann. Probab., Tome 31 (2003) no. 4, pp. 1917-1934 | Article

[33] Saloff-Coste, Laurent; Sheng, Tianyi Random walks and isoperimetric profiles under moment conditions (https://arxiv.org/abs/1501.05929 )

[34] Tessera, R. Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, Comment. Math. Helv., Tome 86 (2011) no. 3, pp. 499-535 | Article

[35] Thompson, R. The rate of escape of random walks on polycyclic and metabelian groups, Ann. Inst. Henri Poincaré Probab. Stat., Tome 49 (2013) no. 1, pp. 270-287 | Article

[36] Valette, A. Nouvelles approches de la propriété (T) de Kazhdan, Société Mathématique de France, Astérisque, Tome 294 (2004), vii+97–124 pages

[37] Varopoulos, N. T. Long range estimates for Markov chains, Bull. Sci. Math., Tome 139 (1985) no. 3, pp. 225-252

[38] Woess, Wolfgang Random Walks on Infinite Graphs and Groups, Cambridge University Press, Cambridge tracts in mathematics, Tome 138 (200), xi+344 pages