Results 1  10
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24
Spectral computations on lamplighter groups and DiestelLeader graphs
, 2004
"... The DiestelLeader graph DL(q, r) is the horocyclic product of the homogeneous trees with respective degrees q+1 and r+1. When q = r, it is the Cayley graph of the lamplighter group (wreath product) Zq≀Z with respect to a natural generating set. For the “Simple random walk ” (SRW) operator on the la ..."
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Cited by 37 (19 self)
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The DiestelLeader graph DL(q, r) is the horocyclic product of the homogeneous trees with respective degrees q+1 and r+1. When q = r, it is the Cayley graph of the lamplighter group (wreath product) Zq≀Z with respect to a natural generating set. For the “Simple random walk ” (SRW) operator on the latter group, Grigorchuk and ˙ Zuk and Dicks and Schick have determined the spectrum and the (ondiagonal) spectral measure (Plancherel measure). Here, we show that thanks to the geometric realization, these results can be obtained for all DLgraphs by directly computing an ℓ 2complete orthonormal system of finitely supported eigenfunctions of the SRW. This allows computation of all matrix elements of the spectral resolution, including the Plancherel measure. As one application, we determine the sharp asymptotic behaviour of the Nstep return probabilities of SRW. The spectral computations involve a natural approximating sequence of finite subgraphs, and we study the question whether the cumulative spectral distributions of the latter converge weakly to the Plancherel measure. To this end, we provide a general result regarding Følner approximations; in the specific case of DL(q, r), the answer is positive only when r = q.
Amenability via random walks
 Duke Math. J
, 2005
"... We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and ˙ Zuk. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are sepa ..."
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Cited by 10 (2 self)
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We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and ˙ Zuk. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups. 1
functions on selfsimilar graphs and bounds for the spectrum
 of the Laplacian, Ann. Inst. Fourier (Grenoble
"... Selfsimilar graphs can be seen as discrete versions of fractals (more precisely: compact, complete metric spaces defined as the fixed set of an iterated system of contractions, see Hutchinson in [15]). The simple random walk is a crucial tool in order to study diffusion on fractals, see Barlow and ..."
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Cited by 9 (1 self)
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Selfsimilar graphs can be seen as discrete versions of fractals (more precisely: compact, complete metric spaces defined as the fixed set of an iterated system of contractions, see Hutchinson in [15]). The simple random walk is a crucial tool in order to study diffusion on fractals, see Barlow and
ON THE SPECTRUM OF LAMPLIGHTER GROUPS AND PERCOLATION CLUSTERS
, 712
"... Abstract. Let G be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let H be a finite group and H ≀ G the lamplighter group (wreath product) over G with group of “lamps ” H. We show that the spectral measure (Plancherel measure) of ..."
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Cited by 6 (1 self)
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Abstract. Let G be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let H be a finite group and H ≀ G the lamplighter group (wreath product) over G with group of “lamps ” H. We show that the spectral measure (Plancherel measure) of any symmetric “switch–walk–switch ” random walk on H ≀G coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on X with parameter p = 1/H. The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilites on the percolation cluster. In particular, if the clusters of percolation with parameter p are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk and ˙ Zuk, resp. Dicks and Schick regarding the case when G is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated density of states of site (or bond) percolation with arbitrary parameter p is always related with the Plancherel measure of a convolution operator by a signed measure on H ≀ G, where H = Z or another suitable group. 1.
Finitely Ramified Iterated Extensions
"... Let p be a prime number, K a number field, and S a finite set of places of K. LetKSbe the compositum of all extensions of K (in a fixed algebraic closure K) which are unramified ..."
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Cited by 4 (0 self)
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Let p be a prime number, K a number field, and S a finite set of places of K. LetKSbe the compositum of all extensions of K (in a fixed algebraic closure K) which are unramified
On the growth of iterated monodromy groups
, 2004
"... The iteration of a quadratic polynomial f = fc(z) = z 2 + c describes a dynamical system in�. The behavior of this system is ruled by the geometry of the orbit O = Oc: = { f(0),f(f(0)) ,..., f i (0),...} of its unique critical point [Miln99]. V. Nekrashevych [Nekr03] associates to each such system ..."
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Cited by 3 (0 self)
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The iteration of a quadratic polynomial f = fc(z) = z 2 + c describes a dynamical system in�. The behavior of this system is ruled by the geometry of the orbit O = Oc: = { f(0),f(f(0)) ,..., f i (0),...} of its unique critical point [Miln99]. V. Nekrashevych [Nekr03] associates to each such system a group of automorphisms of the infinite binary rooted tree T (2). In Section 1, we will sketch the construction of this group known as the iterated monodromy group of f denoted by IMG (f). This note addresses the following conjecture: Conjecture 1 (Nekrashevych). Suppose the critical orbit of f is postcritically finite, i.e., the orbit of the critical point 0 is finite and does not contain 0. Then IMG (f) has intermediate growth. We want to illustrate some of the difficulties that arise in attacking this conjecture. Our plan is to present three examples G, H and I of finitely generated subgroups of Aut(T (2)), all of which have subexponential growth. The group G is the First Grigorchuk group; it was the first known example of a group of intermediate growth [Grig83]. The group H belongs to the family of groups of intermediate growth studied in [Grig84]. Our proofs of subexponential growth for G and H are designed to illustrate the use of Proposition 10. We apply these ideas to prove subexponential growth on I: = IMG (z 2 + i), thus providing the first nontrivial example to support Conjecture
THE SPECTRAL PROBLEM, SUBSTITUTIONS AND ITERATED MONODROMY
, 2007
"... Abstract. We provide a selfsimilar measure for the selfsimilar groupG acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial z 2 + i. We also provide an Lpresentation for G and calculations related to the spectrum of the Markov operator on ..."
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Cited by 2 (2 self)
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Abstract. We provide a selfsimilar measure for the selfsimilar groupG acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial z 2 + i. We also provide an Lpresentation for G and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of G on the orbit of a point on the boundary of the binary rooted tree.
THE AUTOMORPHISM TOWER OF GROUPS ACTING ON ROOTED TREES
, 2003
"... Abstract. The group of isometries Aut(Tn) of a rooted nary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in Aut(Tn). This fact has stimulated the computation of the group of automorphisms of such wellknown examples as the group G stud ..."
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Cited by 2 (1 self)
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Abstract. The group of isometries Aut(Tn) of a rooted nary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in Aut(Tn). This fact has stimulated the computation of the group of automorphisms of such wellknown examples as the group G studied by R. Grigorchuk, and the group ¨ Γ studied by N. Gupta and the second author. In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as G and ¨ Γ. We describe this tower for all subgroups of Aut(T2) which decompose as infinitely iterated wreath products. Furthermore, we describe fully the towers of G and ¨ Γ. More precisely, the tower of G is infinite countable, and the terms of the tower are 2groups. Quotients of successive terms are infinite elementary abelian 2groups. In contrast, the tower of ¨ Γ has length 2, and its terms are {2, 3}groups. We show that Aut 2 ( ¨ Γ)/Aut ( ¨ Γ) is an elementary abelian 3group of countably infinite rank, while Aut 3 ( ¨ Γ) = Aut 2 ( ¨ Γ). 1.