No abstract

No abstract

Abstract. Let T 1 , . . . , T d be homogeneous trees with degrees q 1 +1, . . . , q d +1 ≥ 3, respectively. For each tree, let h : T j → Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product ofequipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d = 2 and q 1 = q 2 = q then we obtain a Cayley graph of the lamplighter group (wreath product) Z q Z. If d = 3 and q 1 = q 2 = q 3 = q then DL is a Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. In general, when d ≥ 4 and q 1 = · · · = q d = q is such that each prime power in the decomposition of q is larger than d − 1, we show that DL is a Cayley graph of a finitely presented group. This group is of type F d−1 , but not F d . It is not automatic, but it is an automata group in most cases. On the other hand, when the q j do not all coincide, DL(q 1 , . . . , q d ) is a vertextransitive graph, but is not a Cayley graph of a finitely generated group. Indeed, it does not even admit a group action with finitely many orbits and finite point stabilizers. The 2 -spectrum of the "simple random walk" operator on DL is always pure point. When d = 2, it is known explicitly from previous work, while for d = 3 we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on DL. It coincides with a part of the geometric boundary of DL.

No abstract

L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/