JournalsjemsVol. 10 , No. 3DOI 10.4171/jems/130

Horocyclic products of trees

  • Wolfgang Woess

    Technische Universität Graz, Austria
  • Markus Neuhauser

    Universität Wien, Austria
  • Laurent Bartholdi

    Georg-August-Universität Göttingen, Germany
Horocyclic products of trees cover


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.