Horocyclic products of trees

  • Wolfgang Woess

    Technische Universität Graz, Austria
  • Markus Neuhauser

    Universität Wien, Austria
  • Laurent Bartholdi

    Georg-August-Universität Göttingen, Germany


Let T1,,TdT_1,\dots, T_d be homogeneous trees with degrees q1+1,,qd+13,q_1+1, \dots, q_d+1 \ge 3, respectively. For each tree, let \hor:TjZ\hor:T_j \to \Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T1,,TdT_1,\dots, T_d is the graph \DL(q1,,qd)\DL(q_1,\dots,q_d) consisting of all dd-tuples x1xdT1××Tdx_1 \cdots x_d \in T_1 \times \dots \times T_d with \hor(x1)++\hor(xd)=0\hor(x_1)+\dots+\hor(x_d)=0, equipped 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=2d=2 and q1=q2=qq_1=q_2=q then we obtain a Cayley graph of the lamplighter group (wreath product) \ZqZ\Zq \wr \Z. If d=3d = 3 and q1=q2=q3=qq_1 = q_2 = q_3 = q then \DL\DL is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when d4d\ge 4 and q1==qd=qq_1 = \dots = q_d = q is such that each prime power in the decomposition of qq is larger than d1d-1, we show that \DL\DL is a Cayley graph of a finitely presented group. This group is of type Fd1F_{d-1}, but not FdF_d. It is not automatic, but it is an automata group in most cases. On the other hand, when the qjq_j do not all coincide, \DL(q1,,qd)\DL(q_1,\dots,q_d) is a vertex-transitive graph, but is not the 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\ell^2-spectrum of the ``simple random walk'' operator on \DL\DL is always pure point. When d=2d=2, it is known explicitly from previous work, while for d=3d=3 we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on \DL\DL. It coincides with a part of the geometric boundary of \DL\DL.

Cite this article

Wolfgang Woess, Markus Neuhauser, Laurent Bartholdi, Horocyclic products of trees. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 771–816

DOI 10.4171/JEMS/130