Metabelian groups with large return probability

Jacoboni, Lison

Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 2121-2167.

Dans cet article, on s’intéresse au comportement asymptotique de la probabilité de retour de la marche aléatoire dans un groupe métabélien de type fini. Pour de tels groupes à croissance exponentielle, on obtient une caractérisation de ceux dont la probabilité de retour est la plus grande en des termes purement algébriques, à l’aide de la dimension de Krull du groupe. Nous obtenons d’abord des bornes inférieures dépendant de la dimension de Krull sur la probabilité de retour des groupes métabéliens dont le sous-groupe dérivé est de torsion. On prouve également une variante respectant la dimension de Krull d’un théorème de Kaloujinine et Krasner pour les groupes métabéliens. Enfin, on étudie des sections particulières de ces groupes afin de donner des bornes supérieures sur la probabilité de retour en fonction de la dimension de Krull.

We investigate the asymptotic behaviour of the return probability of the random walk in finitely generated metabelian groups. For such groups with exponential volume growth, we obtain a characterization of metabelian groups whose return probability is the largest in purely algebraic terms, namely the Krull dimension of the group. Along the way, we give lower bounds on the return probability for metabelian groups with torsion derived subgroup, according to the dimension. We also establish a variation of the famous embedding theorem of Kaloujinine and Krasner for metabelian groups that respects the Krull dimension. Finally, we study specific sections of these groups, and use them to give upper bounds on the return probability in terms of the Krull dimension.

Classification : 20F69, 20E15, 20E22, 20P05
Keywords: Krull dimension, metabelian groups, return probability
Mot clés : dimension de Krull, groupes métabéliens, probabilité de retour

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