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2.0

Jun 30, 2018
06/18

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Laurent Bartholdi; Olivier Siegenthaler; Todd Trimble

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We define wreath products of cocommutative Hopf algebras, and show that they enjoy a universal property of classifying cleft extensions, analogous to the Kaloujnine-Krasner theorem for groups. We show that the group ring of a wreath product of groups is the wreath product of their group rings, and that (with a natural definition of wreath products of Lie algebras) the universal enveloping algebra of a wreath product of Lie algebras is the wreath product of their enveloping algebras. We recover...

Topics: Mathematics, Category Theory, Rings and Algebras, Group Theory

Source: http://arxiv.org/abs/1407.3835

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Sep 21, 2013
09/13

by
Laurent Bartholdi; Oleg Bogopolski

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We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. We also construct a finitely generated, torsion-free group which can be mapped onto Z and which has a finitely generated commensurator.

Source: http://arxiv.org/abs/0902.4542v1

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2.0

Jun 28, 2018
06/18

by
Laurent Bartholdi; Dzmitry Dudko

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We develop a general theory of "bisets": sets with two commuting group actions. They naturally encode topological correspondences. Just as van Kampen's theorem decomposes into a graph of groups the fundamental group of a space given with a cover, we prove analogously that the biset of a correspondence decomposes into a "graph of bisets": a graph with bisets at its vertices, given with some natural maps. The "fundamental biset" of the graph of bisets recovers the...

Topics: Dynamical Systems, Group Theory, Mathematics

Source: http://arxiv.org/abs/1512.08539

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5.0

Jun 28, 2018
06/18

by
Laurent Bartholdi

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We consider decidability problems associated with Engel's identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given $x,y$, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements $y$ such that the map $x\mapsto[x,y]$ attracts to $\{1\}$. Although this...

Topics: Formal Languages and Automata Theory, Computing Research Repository, Group Theory, Mathematics

Source: http://arxiv.org/abs/1512.01717

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2.0

Jun 28, 2018
06/18

by
Laurent Bartholdi

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These are introductory notes on word growth of groups, and include a gentle presentation of wreath products as well as recent results on construction of groups of with given growth function. They are are an expanded version of a mini-course given at "Le Louverain", June 24-27, 2014.

Topics: Group Theory, Mathematics

Source: http://arxiv.org/abs/1512.07044

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4.0

Jun 29, 2018
06/18

by
Laurent Bartholdi; Dzmitry Dudko

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We consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as "mapping class bisets". We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called "small mapping class bisets". We phrase the decision problem of "Thurston equivalence" between branched self-coverings of the sphere in terms of the conjugacy and centralizer...

Topics: Group Theory, Logic in Computer Science, Dynamical Systems, Computing Research Repository,...

Source: http://arxiv.org/abs/1603.04059

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Sep 24, 2013
09/13

by
Laurent Bartholdi; Andre G. Henriques; Volodymyr V. Nekrashevych

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We explore the connections between automata, groups, limit spaces of self-similar actions, and tilings. In particular, we show how a group acting ``nicely'' on a tree gives rise to a self-covering of a topological groupoid, and how the group can be reconstructed from the groupoid and its covering. The connection is via finite-state automata. These define decomposition rules, or self-similar tilings, on leaves of the solenoid associated with the covering.

Source: http://arxiv.org/abs/math/0412373v2

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2.0

Jun 29, 2018
06/18

by
Laurent Bartholdi; Dzmitry Dudko

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Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as maps with "contracting biset". We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a...

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1610.02434

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Sep 21, 2013
09/13

by
Laurent Bartholdi; Anna Erschler

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We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in (G,S_i) and in (H,T) coincide. Given a nilpotent group G, we characterize its connected component in this graph: if that connected component contains at least one torsion-free group, then it consists of those groups which generate the same variety of groups as...

Source: http://arxiv.org/abs/1301.4669v1

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Sep 18, 2013
09/13

by
Laurent Bartholdi; Rostislav I. Grigorchuk

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In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an N-series of subgroups. The asymptotics of the Poincare series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial growth and growth of type $e^{\sqrt n}$ in the class of residually-p groups, and gives examples of finitely generated p-groups of uniformly exponential growth. In the second part, we produce two...

Source: http://arxiv.org/abs/math/0002010v2

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Sep 20, 2013
09/13

by
Laurent Bartholdi; Vadim A. Kaimanovich; Volodymyr V. Nekrashevych

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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.

Source: http://arxiv.org/abs/0802.2837v1

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Sep 23, 2013
09/13

by
Laurent Bartholdi

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We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive abscissa of convergence, is algebraic over the ring Q[2^{-s},...,P^{-s}] for some integer P, and show that it can be analytically continued (through root singularities) to the left half-plane. We compute the abscissa of convergence and the functional equation...

Source: http://arxiv.org/abs/1303.1805v1

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4.0

Jun 29, 2018
06/18

by
Laurent Bartholdi; Dawid Kielak

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We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in this manner a characterization of amenability in terms of cellular automata: "A group $G$ is amenable if and only if every cellular automaton with carrier $G$ that has gardens of Eden also has mutually erasable patterns." This answers a question by Schupp, and solves a conjecture by Ceccherini-Silberstein, Mach\`i and Scarabotti. An appendix by Dawid Kielak proves that group rings without zero divisors...

Topics: Group Theory, Formal Languages and Automata Theory, Computing Research Repository, Mathematics

Source: http://arxiv.org/abs/1605.09133

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Sep 21, 2013
09/13

by
Laurent Bartholdi; Anna G. Erschler

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We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all natural number k a torsion group with growth function asymptotically $\exp(n^{1-(1-\alpha)^k}),\quad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2$, and a torsion-free group with growth function asymptotically $\exp(\log(n)n^{1-(1-\alpha)^k})$. These are the first examples of groups of intermediate growth for which the growth...

Source: http://arxiv.org/abs/1011.5266v2

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Sep 21, 2013
09/13

by
Bettina Eick; René Hartung; Laurent Bartholdi

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The main part of this paper contains a description of a nilpotent quotient algorithm for L-presented groups and a report on applications of its implementation in the computer algebra system GAP. The appendix introduces two new infinite series of L-presented groups. Apart from being of interest in their own right, these new L-presented groups serve as examples for applications of the nilpotent quotient algorithm.

Source: http://arxiv.org/abs/0706.3131v2

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Sep 23, 2013
09/13

by
Laurent Bartholdi; Tullio G. Ceccherini-Silberstein

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Consider the tesselation of the hyperbolic plane by m-gons, l per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate "holly trees", a family of reduced loops in these graphs. We then apply Grigorchuk's result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.

Source: http://arxiv.org/abs/math/0109069v2

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Sep 18, 2013
09/13

by
Laurent Bartholdi; Pierre de la Harpe

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Let G be a group which has for all n a finite number r_n(G) of irreducible complex linear representations of dimension n. Let $\zeta(G,s) = \sum_{n=1}^{\infty} r_n(G) n^{-s}$ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for $\zeta(G,s)$ in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in...

Source: http://arxiv.org/abs/0809.0131v2

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Sep 19, 2013
09/13

by
Laurent Bartholdi; Rostislav I. Grigorchuk

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We study the subgroup structure, Hecke algebras, quasi-regular 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 quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition...

Source: http://arxiv.org/abs/math/9911206v2

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Sep 21, 2013
09/13

by
Laurent Bartholdi; Rostislav I. Grigorchuk

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We construct a group acting on a binary rooted tree; this discrete group mimics the monodromy action of iterates of $f(z)=z^2-1$ on associated coverings of the Riemann sphere. We then derive some algebraic properties of the group, and describe for that specific example the connection between group theory, geometry and dynamics. The most striking is probably that the quotient Cayley graphs of the group (aka ``Schreier graphs'') converge to the Julia set of $f$.

Source: http://arxiv.org/abs/math/0203244v1

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Sep 22, 2013
09/13

by
Laurent Bartholdi; Zoran Šunik

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It is shown that certain ascending HNN extensions of free abelian groups of finite rank, as well as various lamplighter groups, can be realized as automaton groups, i.e., can be given a self-similar structure. This includes the solvable Baumslag-Solitar groups BS(1,m), for m not +/-1. In addition, it is shown that, for any relatively prime integers m,n>=2, the pair of Baumslag-Solitar groups BS(1,m) and BS(1,n) can be realized by a pair of dual automata. The examples are then used to...

Source: http://arxiv.org/abs/math/0603032v2

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7.0

Jun 30, 2018
06/18

by
Laurent Bartholdi; Yves de Cornulier; Dessislava Kochloukova

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We study the homological finiteness property FPm of permutational wreath products.

Topics: Mathematics, Group Theory

Source: http://arxiv.org/abs/1406.5261

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Sep 23, 2013
09/13

by
Laurent Bartholdi; Yves de Cornulier

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We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group.

Source: http://arxiv.org/abs/math/0510141v1

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39

Sep 20, 2013
09/13

by
Laurent Bartholdi

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We study amenability of affine algebras (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring KG is amenable for some (and therefore for any) field K.

Source: http://arxiv.org/abs/math/0608302v2

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Sep 22, 2013
09/13

by
Laurent Bartholdi

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I describe a class of groups acting on rooted trees that all have intermediate word growth between polynomial and exponential. The argument constructs a functional equation on the growth formal power series, and derives the growth properties from its analytical behaviour. This places under a unified proof all known or conjectured examples of such groups, and answers a few open questions by Rostislav Grigorchuk.

Source: http://arxiv.org/abs/math/0201293v2

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Sep 20, 2013
09/13

by
Laurent Bartholdi; Serge Cantat; Tullio Ceccherini-Silberstein; Pierre de la Harpe

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Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.

Source: http://arxiv.org/abs/math/0612409v1

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Sep 23, 2013
09/13

by
Laurent Bartholdi; Olivier Siegenthaler

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We study a twisted version of Grigorchuk's first group, and stress its similarities and differences to its model. In particular, we show that it admits a finite endomorphic presentation, has infinite-rank multiplier, and does not have the congruence property.

Source: http://arxiv.org/abs/0904.1600v1

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3.0

Jun 30, 2018
06/18

by
Laurent Bartholdi; Anna Erschler

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Every countable group that does not contain a finitely generated subgroup of exponential growth imbeds in a finitely generated group of subexponential growth. This produces in particular the first examples of groups of subexponential growth containing the additive group of the rationals.

Topics: Mathematics, Functional Analysis, Group Theory

Source: http://arxiv.org/abs/1403.5584

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45

Sep 18, 2013
09/13

by
Laurent Bartholdi

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We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of `bumps' on paths in a graph: in a $d$-regular (not necessarily transitive) non-oriented graph let the series $G(t)$ count all paths between two fixed points weighted by their length $t^{length}$, and $F(u,t)$ count the same paths, weighted as $u^{number of...

Source: http://arxiv.org/abs/math/0012161v2

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45

Sep 18, 2013
09/13

by
Laurent Bartholdi

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We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has finite width while the other, the ``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki. More precisely, the Lie algebra of the Gupta-Sidki group has Gelfand-Kirillov dimension $\log3/\log(1+\sqrt2)$. We then draw a general result relating the growth of a branch group, of its Lie algebra, of its graded group ring, and of a natural homogeneous space we call...

Source: http://arxiv.org/abs/math/0101222v3

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Sep 23, 2013
09/13

by
Laurent Bartholdi; Anna Erschler

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We show that there exists a finitely generated group of growth ~f for all functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq f(\eta R) for all R large enough and \eta\approx2.4675 the positive root of X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than \exp(R^{\log2/\log\eta}). We also give a family of self-similar branched groups of growth ~\exp(R^\alpha) for a dense set of \alpha\in(\log2/\log\eta,1).

Source: http://arxiv.org/abs/1110.3650v3

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129

Jul 20, 2013
07/13

by
Laurent Bartholdi

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I answer a question from the 1993 International Mathematical Olympiads by constructing an equivalent algebraic problem, and unearth a surprising behaviour of some polynomials over the two-element field.

Source: http://arxiv.org/abs/math/9910056v1

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Sep 18, 2013
09/13

by
Laurent Bartholdi; Fabrice Liardet

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We describe the links between group theory and psychology, in particular through the works of Piaget. We show that groups appear universally in his description of children's intelligence, and that the notion of groupoid, which was little considered in psychology, may be fundamental. We study in particular the applicability of group theory concepts to the development of educative games.

Source: http://arxiv.org/abs/math/0505651v1

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Sep 23, 2013
09/13

by
Laurent Bartholdi; Pedro V. Silva

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This text, Chapter 23 in the "AutoMathA" handbook, is devoted to the study of rational subsets of groups, with particular emphasis on the automata-theoretic approach to finitely generated subgroups of free groups. Indeed, Stallings' construction, associating a finite inverse automaton with every such subgroup, inaugurated a complete rewriting of free group algorithmics, with connections to other fields such as topology or dynamics. Another important vector in the chapter is the...

Source: http://arxiv.org/abs/1012.1532v1

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106

Sep 23, 2013
09/13

by
Laurent Bartholdi; Rostislav I. Grigorchuk; Zoran Sunik

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This is a long introduction to the theory of "branch groups": groups acting on rooted trees which exhibit some self-similarity features in their lattice of subgroups.

Source: http://arxiv.org/abs/math/0510294v2

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Sep 18, 2013
09/13

by
Laurent Bartholdi; Rostislav I. Grigorchuk

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We study spectra of noncommutative dynamical systems, representations of fractal groups, and regular graphs. We explicitly compute these spectra for five examples of groups acting on rooted trees, and in three cases obtain totally disconnected sets.

Source: http://arxiv.org/abs/math/0012174v1

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Jul 20, 2013
07/13

by
Laurent Bartholdi; Rostislav I. Grigorchuk

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We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite approximation'' method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also ``substitutional graphs''. We also formulate our results in terms of Hecke type operators related to some...

Source: http://arxiv.org/abs/math/9910102v1

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Sep 20, 2013
09/13

by
Laurent Bartholdi

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We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field k we construct a k-algebra K which (1) is finitely generated and infinite-dimensional, but has only finite-dimensional quotients; (2) has a subalgebra of finite codimension, isomorphic to $M_2(K)$; (3) is prime; (4)...

Source: http://arxiv.org/abs/math/0410226v3

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Sep 20, 2013
09/13

by
Laurent Bartholdi; Illya I. Reznykov

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We consider a very simple Mealy machine (three states over a two-symbol alphabet), and derive some properties of the semigroup it generates. In particular, this is an infinite, finitely generated semigroup; we show that the growth function of its balls behaves asymptotically like n^2.4401..., where this constant is 1 + log(2)/log((1+sqrt(5))/2); that the semigroup satisfies the identity g^6=g^4; and that its lattice of two-sided ideals is a chain.

Source: http://arxiv.org/abs/math/0506203v2

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Sep 18, 2013
09/13

by
Laurent Bartholdi; Benjamin Enriquez; Pavel Etingof; Eric Rains

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For a positive integer n we introduce quadratic Lie algebras tr_n qtr_n and discrete groups Tr_n, QTr_n naturally associated with the classical and quantum Yang-Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras tr_n, qtr_n are Koszul, and find their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). We construct cell complexes which are classifying...

Source: http://arxiv.org/abs/math/0509661v6

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Sep 23, 2013
09/13

by
Laurent Bartholdi; Volodymyr Nekrashevych

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We answer Hubbard's question on determining the Thurston equivalence class of ``twisted rabbits'', i.e. images of the ``rabbit'' polynomial under n-th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.

Source: http://arxiv.org/abs/math/0510082v4

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Sep 21, 2013
09/13

by
Laurent Bartholdi; Olivier Siegenthaler; Pavel Zalesskii

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We state and study the congruence subgroup problem for groups acting on rooted tree, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is Abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels. We work out explicitly known and new examples of...

Source: http://arxiv.org/abs/0902.3220v1

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Sep 23, 2013
09/13

by
Laurent Bartholdi; André G. Henriques

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There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral....

Source: http://arxiv.org/abs/1202.3033v1

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Jul 20, 2013
07/13

by
Laurent Bartholdi

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In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth $\gamma$: $e^{\sqrt{n}}\precsim\gamma(n)\precsim e^{n^\beta},$ where $\beta=\log_{32}(31)\approx0.991$. He conjectured that the lower bound is actually tight. In this paper we improve the lower bound to $e^{n^\alpha}\precsim\gamma(n),$ where $\alpha\approx0.5157$, with the aid of a computer. This disproves the conjecture that the lower bound be tight.

Source: http://arxiv.org/abs/math/9910068v1

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Sep 17, 2013
09/13

by
Laurent Bartholdi

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We give a general definition of self-similar Lie algebras, and show that important examples of Lie algebras fall into that class. We give sufficient conditions for a self-similar Lie algebra to be nil, and prove in this manner that the self-similar algebras associated with Grigorchuk's and Gupta-Sidki's torsion groups are nil as well as self-similar. We derive the same results for a class of examples constructed by Petrogradsky, Shestakov and Zelmanov.

Source: http://arxiv.org/abs/1003.1125v1

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Sep 18, 2013
09/13

by
Agata Smoktunowicz; Laurent Bartholdi

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For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl.

Source: http://arxiv.org/abs/1102.2697v1

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Jul 20, 2013
07/13

by
Laurent Bartholdi

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We describe, up to degree equal to the rank, the Lie algebra associated with the automorphism group of a free group. We compute in particular the ranks of its homogeneous components, and their structure as modules over the linear group. Along the way, we infirm (but confirm a weaker form of) a conjecture by Andreadakis, and answer a question by Bryant-Gupta-Levin-Mochizuki.

Source: http://arxiv.org/abs/1304.0498v2

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Sep 18, 2013
09/13

by
Laurent Bartholdi; Zoran Sunik

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We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional condition, they are periodic).

Source: http://arxiv.org/abs/math/0005113v2

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Sep 18, 2013
09/13

by
Laurent Bartholdi

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This note constructs a finitely generated group $W$ whose word-growth is exponential, but for which the infimum of the growth rates over all finite generating sets is 1 -- in other words, of non-uniformly exponential growth. This answers a question by Mikhael Gromov. The construction also yields a group of intermediate growth $V$ that locally resembles $W$ in that (by changing the generating set of $W$) there are isomorphic balls of arbitrarily large radius in $V$ and $W$'s Cayley graphs.

Source: http://arxiv.org/abs/math/0210471v1

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Jul 20, 2013
07/13

by
Laurent Bartholdi

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We show that contracting self-similar groups satisfy the Farrell-Jones conjectures as soon as their universal contracting cover is non-positively curved. This applies in particular to bounded self-similar groups. We define, along the way, a general notion of contraction for groups acting on a rooted tree in a not necessarily self-similar manner.

Source: http://arxiv.org/abs/1107.5339v2

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Sep 17, 2013
09/13

by
Laurent Bartholdi

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We introduce L-presentations: group presentations given by a generating set, a set of relations and a set of substitution rules on the generating set producing more relations. We first study in full generality the structure of finitely L-presented groups, i.e. groups for which all the above sets are finite, and then show that a broad class of groups acting on rooted trees admit explicitly constructible finite L-presentations: they are the "branch" groups defined by R. Grigorchuk.

Source: http://arxiv.org/abs/math/0007062v3