Critical constants for recurrence of random walks on $G$-spaces

Erschler, Anna

Annales de l'Institut Fourier, Tome 55 (2005) no. 2, pp. 493-509.

On introduit la notion de constante critique $c_{rt}$ pour les marches aléatoires sur les $G$-espaces. Pour un sous-groupe $H$ dans un groupe de type fini $G$, la constante critique de la récurrence est un invariant asymptotique du $G$-espace $G/H$. On montre que pour chaque $G$-espace infini, $c_{rt}\ge 1/2$. On dit que $G/H$ est très petit si $c_{rt}<1$. Pour un sous-groupe distingué $H$ l’espace quotient $G/H$ est très petit si et seulement si il est fini. Cependant, on donne des exemples de $G$-espaces très petits et infinis. On montre également que la constante critique pour la récurrence peut être utilisée pour estimer la croissance de groupes et la vitesse de fuite des marches aléatoires.

We introduce the notion of a critical constant $c_{rt}$ for recurrence of random walks on $G$-spaces. For a subgroup $H$ of a finitely generated group $G$ the critical constant is an asymptotic invariant of the quotient $G$-space $G/H$. We show that for any infinite $G$-space $c_{rt}\ge 1/2$. We say that $G/H$ is very small if $c_{rt}<1$. For a normal subgroup $H$ the quotient space $G/H$ is very small if and only if it is finite. However, we give examples of infinite very small $G$-spaces. We show also that critical constants for recurrence can be used to estimate the growth of groups as well as the drift for random walks on groups.

Classification : 20F65, 20E08, 60B15
Keywords: growth of groups, Grigorchuk groups, branch groups, random walks, recurrence, drift
Mot clés : croissance des groupes, groupes de Grigorchuk, groupes branches, marches aléatoires, récurrence, vitesse de fuite.

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