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000160775 001__ 160775 000160775 005__ 20181203022215.0 000160775 0247_ $$2doi$$a10.4171/JEMS/130 000160775 02470 $$2ISI$$a000257869200008 000160775 037__ $$aARTICLE 000160775 245__ $$aHorocyclic products of trees 000160775 269__ $$a2008 000160775 260__ $$c2008 000160775 336__ $$aJournal Articles 000160775 520__ $$aLet T-1, ... , T-d be homogeneous trees with degrees q(1) + 1, ... , q(d) + 1 >= 3; respectively. For each tree, let h : Tj -> 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(q1, ... , q(d)) consisting of all d-tuples x(1) ... x(d) is an element of T-1 x ... x T-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) 3q (sic) 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 Fd - 1, but not Fd. 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 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 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. 000160775 6531_ $$arestricted wreath product 000160775 6531_ $$atrees 000160775 6531_ $$ahorocycles 000160775 6531_ $$aDiestel-Leader graph 000160775 6531_ $$agrowth function 000160775 6531_ $$anormal form 000160775 6531_ $$aMarkov operator 000160775 6531_ $$aspectrum 000160775 6531_ $$aDiestel-Leader Graphs 000160775 6531_ $$aFinitely Presented Group 000160775 6531_ $$aArc-Transitive Digraphs 000160775 6531_ $$aRandom-Walks 000160775 6531_ $$aEquilateral Triangle 000160775 6531_ $$aLamplighter Groups 000160775 6531_ $$aInfinite-Graphs 000160775 6531_ $$aAffine Group 000160775 6531_ $$aBoundary 000160775 700__ $$aBartholdi, Laurent 000160775 700__ $$aNeuhauser, Markus 000160775 700__ $$aWoess, Wolfgang 000160775 773__ $$j10$$tJournal Of The European Mathematical Society$$q771-816 000160775 909C0 $$xU10077$$0252369$$pSB 000160775 909CO $$pSB$$particle$$ooai:infoscience.tind.io:160775 000160775 917Z8 $$xWOS-2010-11-30 000160775 937__ $$aEPFL-ARTICLE-160775 000160775 973__ $$rREVIEWED$$sPUBLISHED$$aEPFL 000160775 980__ $$aARTICLE