Random walks on free products
Annales de l'Institut Fourier, Volume 41 (1991) no. 2, p. 467-491

Let G=* j=1 q+1 G n j +1 be the product of q+1 finite groups each having order n j +1 and let μ be the probability measure which takes the value p j /n j on each element of G n j +1 {e}. In this paper we shall describe the point spectrum of μ in C reg * (G) and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers n j . We also compute the continuous spectrum of μ in C reg * (G) in several cases. A family of irreducible representations of G, parametrized on the continuous spectrum of μ, is here presented. Finally, we shall get a decomposition of the regular representation of G by means of the Green function of μ and the decomposition is into irreducibles if and only if there are no true eigenspaces for μ.

Soit G=* j=1 q+1 G n j +1 le produit libre de q+1 groupes finis d’ordre n j +1, et μ la probabilité prenant la valeur p j /n j sur chaque élément de G n j +1 {e}. Nous décrivons ici le spectre ponctuel de μ sur C reg * (G). On montre en particulier que ce spectre ponctuel apparaît pour certains choix des nombres n j , et les espaces propres correspondants dans l 2 sont décrits. Enfin, on obtient une décomposition de la représentation régulière de G à l’aide de la fonction de Green de μ, cette décomposition étant irréductible si, et seulement si, μ n’a pas de sous-espace propre.

@article{AIF_1991__41_2_467_0,
     author = {Kuhn, M. Gabriella},
     title = {Random walks on free products},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {41},
     number = {2},
     year = {1991},
     pages = {467-491},
     doi = {10.5802/aif.1261},
     zbl = {0725.60009},
     mrnumber = {93a:43008},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1991__41_2_467_0}
}
Kuhn, M. Gabriella. Random walks on free products. Annales de l'Institut Fourier, Volume 41 (1991) no. 2, pp. 467-491. doi : 10.5802/aif.1261. http://www.numdam.org/item/AIF_1991__41_2_467_0/

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