no. 6
The Liouville property and Hilbertian compression
[Propriété de Liouville et compression Hilbertienne]
Gournay, Antoine1
1 Université de Neuchâtel Institut de mathématiques Rue É.-Argand 11, 2000 Neuchâtel, Switzerland
Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2435-2454.

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 α≥1-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.

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 α≥1-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.

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DOI : 10.5802/aif.3067
Classification : 20F69, 60J50, 60G50, 37A35, 43A65
Keywords: Hilbertian compression, random walks on groups, entropy, drift, growth of groups
Mot clés : compression hilbertienne, marches aléatoires sur les groupes, entropie, vitesse, croissance des groupes

Gournay, Antoine 1

1 Université de Neuchâtel Institut de mathématiques Rue É.-Argand 11, 2000 Neuchâtel, Switzerland 
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Gournay, Antoine. The Liouville property and Hilbertian compression. Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2435-2454. doi : 10.5802/aif.3067. https://aif.centre-mersenne.org/articles/10.5802/aif.3067/

[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., Volume 81 (2006) no. 4, pp. 911-929 | DOI

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

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

[6] Avez, A. Entropie des groupes de type fini, C. R. Acad. Sci. Paris Sér. A-B, Volume 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, Volume 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) (London Math. Soc. Lecture Note Ser.), Volume 227, Cambridge Univ. Press, Cambridge, 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., Volume 354 (2012), pp. 43-72 | DOI

[11] Benyamini, Y.; Lindenstrauss, J. Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48, American Mathematical Society, providence, 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., Volume 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., Volume 89 (1997) no. 1, pp. 133-199 | DOI

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

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

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

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

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

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

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

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

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

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

[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., Volume 41 (2013) no. 5, pp. 3392-3419 | DOI

[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., Volume 157 (2011) no. 1, pp. 53-108 | DOI

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

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

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

[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., Volume 86 (2011) no. 3, pp. 499-535 | DOI

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

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

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

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

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