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51
Lpresentations and branch groups
, 2002
"... Abstract. We introduce Lpresentations: group presentations endowed with a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations. 1. ..."
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Cited by 43 (11 self)
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Abstract. We introduce Lpresentations: group presentations endowed with a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations. 1.
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 35 (14 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.
On parabolic subgroups and Hecke algebras of some fractal groups
, 2001
"... We study the subgroup structure, Hecke algebras, quasiregular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasiregular representations are irreducible. These (inf ..."
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Cited by 27 (12 self)
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We study the subgroup structure, Hecke algebras, quasiregular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasiregular representations are irreducible. These (infinitedimensional) representations are approximated by finitedimensional quasiregular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasiregular representations is given by the double cosets of the parabolic subgroup. Since our results derive from considerations on finiteindex subgroups, they also hold for the profinite completions ̂ G of the groups G. The representations involved have interesting spectral properties investigated in [BG00b]. This paper serves as a grouptheoretic counterpart to the studies in the mentioned paper. We study more carefully a few examples of fractal groups, and in doing so exhibit the first example of a torsionfree branch justinfinite group. We also produce a new example of branch justinfinite group of intermediate growth, and provide for it an Ltype presentation by generators and relators.
ON AMENABILITY OF AUTOMATA GROUPS
, 2008
"... We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (“Mother group ..."
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Cited by 22 (1 self)
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We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (“Mother groups”) which is done by analyzing the asymptotic properties of random walks on these groups.
Asymptotic aspects of Schreier graphs and Hanoi Towers groups
 COMPTES RENDUS MATHÉMATIQUE, ACADÉMIE DES SCIENCES PARIS 342 (2006
, 2006
"... We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of th ..."
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Cited by 15 (4 self)
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We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior.
Electrical networks, symplectic reductions, and applications to the renormalization map of selfsimilar lattices
 PROC. OF SYMP. IN PURE MATH., MANDELBROT JUBILEE. ARXIV/MATHPH/0304015
, 2004
"... The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the “trace map ” and the “gluing map”) and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductio ..."
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Cited by 12 (2 self)
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The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the “trace map ” and the “gluing map”) and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductions, in particular we study the singularities of symplectic reductions when considered as rational maps on Lagrangian Grassmannians. This is motivated by [23] where a renormalization map was introduced in order to describe the spectral properties of selfsimilar lattices. In this text, we show that this renormalization map can be expressed in terms of symplectic reductions and that some of its key properties are direct consequences of general properties of symplectic reductions (and the singularities of the symplectic reduction play an important role in relation with the spectral properties of our operator). We also present new examples where we can compute the renormalization map.
Iterated Monodromy Groups
, 2008
"... We associate a group IMG(f) to every covering f of a topological space M by its open subset. It is the quotient of the fundamental group π1(M) by the intersection of the kernels of its monodromy action for the iterates f n. Every iterated monodromy group comes together with a naturally defined actio ..."
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Cited by 10 (3 self)
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We associate a group IMG(f) to every covering f of a topological space M by its open subset. It is the quotient of the fundamental group π1(M) by the intersection of the kernels of its monodromy action for the iterates f n. Every iterated monodromy group comes together with a naturally defined action on a rooted tree. We present an effective method to compute this action and show how the dynamics of f is related to the group. In particular, the Julia set of f can be reconstructed from IMG(f) (from its action on the tree), if f is expanding. 1
Enumeration problems for classes of selfsimilar graphs
 MR MR2353122
"... Abstract. We describe a general construction principle for a class of selfsimilar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples i ..."
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Cited by 9 (6 self)
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Abstract. We describe a general construction principle for a class of selfsimilar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples involving classical selfsimilar graphs such as the Sierpiński graphs. The enumeration problems we investigate include counting independent subsets, matchings and connected subsets. 1.