no. 2
Poisson boundary of triangular matrices in a number field
Schapira, Bruno1
1 Université Paris-Sud Département de Mathématiques Bât. 425 91405 Orsay Cedex (France)
Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 575-593.

The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio concerning the Poisson boundary of random rational affinities.

L’objet de cette note est de décrire la frontière de Poisson du groupe des matrices triangulaires supérieures inversibles à coefficients dans un corps de nombre. C’est une généralisation en dimension supérieure d’un résultat de Brofferio concernant la frontière de Poisson du groupe des applications affines rationnelles.

DOI: 10.5802/aif.2441
Classification: 22D40, 28D05, 28D20, 60B15, 60J10, 60J50
Keywords: Random walks, Poisson boundary, triangular matrices, number field, Bruhat decomposition
Schapira, Bruno 1

1 Université Paris-Sud Département de Mathématiques Bât. 425 91405 Orsay Cedex (France) 
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Schapira, Bruno. Poisson boundary of triangular matrices in a number field. Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 575-593. doi : 10.5802/aif.2441. https://aif.centre-mersenne.org/articles/10.5802/aif.2441/

[1] Bader, U.; Shalom, Y. Factor and normal subgroup theorems for lattices in products of groups, Invent. Math., Volume 163 (2006), pp. 415-454 | DOI | MR | Zbl

[2] Brofferio, S. The Poisson Boundary of random rational affinities, Ann. Inst. Fourier, Volume 56 (2006), pp. 499-515 | DOI | Numdam | MR | Zbl

[3] Cartwright, D. I.; Kaimanovich, V. A.; Woess, W. Random walks on the affine group of local fields and of homogeneous trees, Ann. Inst. Fourier, Volume 44 (1994), pp. 1243-1288 | DOI | Numdam | MR | Zbl

[4] Derriennic, Y. Entropie, théorèmes limite et marches aléatoires, Probability measures on groups VIII (Lect. Notes Math.), Volume 1210 (1986), pp. 241-284 | MR | Zbl

[5] Élie, L. Noyaux potentiels associés aux marches aléatoires sur les espaces homogènes. Quelques exemples clefs dont le groupe affine, Théorie du potentiel (Lectures Notes in Math.), Volume 1096 (1984), pp. 223-260 | MR | Zbl

[6] Furman, A. Random walks on groups and random transformations, Handbook of dynamical systems, Volume 1A (2002), pp. 931-1014 | MR | Zbl

[7] Furstenberg, H. A Poisson formula for semi-simple Lie groups, Ann. of Math., Volume 77 (1963), pp. 335-386 | DOI | MR | Zbl

[8] Furstenberg, H. Boundary theory and stochastic processes on homogeneous spaces, Harmonic analysis on homogeneous spaces (1973), pp. 193-229 | MR | Zbl

[9] Guivarc’h, Y.; Raugi, A. Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrsch. Verw. Gebiete, Volume 69 (1985), pp. 187-242 | DOI | MR | Zbl

[10] Kaimanovich, V. A. The Poisson formula for groups with hyperbolic properties, Ann. of Math., Volume 152 (2000) no. 2, pp. 659-692 | DOI | MR | Zbl

[11] Lang, S. Introduction to diophantine approximations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966 | MR | Zbl

[12] Ledrappier, F. Poisson boundaries of discrete groups of matrices, Israel J. Math., Volume 50 (1985), pp. 319-336 | DOI | MR | Zbl

[13] Raugi, A. Fonctions harmoniques sur les groupes localement compacts à base dénombrable, Bull. Soc. Math. France Mém. (1977) no. 54, pp. 5-118 | Numdam | MR | Zbl

[14] Samuel, P. Théorie algébrique des nombres, Hermann, Paris, 1967 (130 pp.) | MR | Zbl

[15] Serre, J.-P. Corps locaux, Sec. edition, Publications of University Nancago, Hermann, Paris, 1968 no. VIII (245 pp.) | MR | Zbl

[16] Warner, G. Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, 188, Springer-Verlag, New York-Heidelberg, 1972 (xvi+529 pp.) | MR | Zbl

[17] Weil, A. Basic number theory, Third edition, Die Grundlehren der Mathematischen Wissenschaften, 144, Springer-Verlag, New York-Berlin, 1974 (xviii+325 pp.) | MR | Zbl

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