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      Visual boundaries of Diestel-Leader graphs

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          Abstract

          Diestel-Leader graphs are neither hyperbolic nor CAT(0), so their visual boundaries may be pathological. Indeed, we show that for \(d>2\), \(\partial\text{DL}_d(q)\) carries the indiscrete topology. On the other hand, \(\partial\text{DL}_2(q)\), while not Hausdorff, is \(T_1\), totally disconnected, and compact. Since \(\text{DL}_2(q)\) is a Cayley graph of the lamplighter group \(L_q\), we also obtain a nice description of \(\partial\text{DL}_2(q)\) in terms of the lamp stand model of \(L_q\) and discuss the dynamics of the action.

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          Spaces with nonpositive curvature and their ideal boundaries

          Christopher Croke, Bruce Kleiner (2000)
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            Horocyclic products of trees

            Laurent Bartholdi, Markus Neuhäuser, Wolfgang Woess (2006)
            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 of T_1,...,T_d is the graph DL(q_1,...,q_d) consisting of all d-tuples x_1...x_d in T_1x...xT_d with h(x_1)+...+h(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=2 and q_1=q_2=q then we obtain a Cayley graph of the lamplighter group (wreath product) (Z/qZ) wr Z. If d=3 and q_1=q_2=q_3=q then DL is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also 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 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 l^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.
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              Growth series of some wreath products

              Walter R. Parry (1992)
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                Author and article information

                Journal
                Publication date Created: 08 July 2013
                Publication date Updated: 2015-05-28
                Article
                ArXiV ID: 1307.2163
                SO-VID: dd32cbc2-d94d-4b4b-9cb9-7612c7a8a0c5
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
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                ACM classes 20F65, 20F69, Secondary 20E22, 05C25
                Journal reference Topology Proc. 46:181-204, 2015
                Comments 19 pages, 5 figures The substantive changes made in this revision are mostly found in Section 5.1, where we prove that geodesics in $\dl_d(q)$ have at most one "turn"
                Categories math.GR math.GT

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