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Laurent Bartholdi

Publications and source records attributed to Laurent Bartholdi.

At least 109 records · Page 6Linked to original sources
arXiv2005

Branch Rings, Thinned Rings, Tree Enveloping Rings

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) has quadratic growth, and therefore Gelfand-Kirillov dimension 2; (5) is recursively presented; (6) satisfies no identity; (7) contains a transcendental, invertible element; (8) is semiprimitive if k has characteristic $\neq2$; (9) is graded if k has characteristic 2; (10) is primitive if k is a non-algebraic extension of GF(2); (11) is graded nil and Jacobson radical if k is an algebraic extension of GF(2).

math.RA↗
arXiv2005

Theorie des groupes et psychologie de l'intelligence

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.

math.HO↗
arXiv2004

The smallest Mealy automaton of intermediate growth

In this paper we study the smallest Mealy automaton of intermediate growth, first considered by the last two authors. We describe the automatic transformation monoid it defines, give a formula for the generating series for its (ball volume) growth function, and give sharp asymptotics for its growth function, namely [ F(n) \sim 2^{5/2} 3^{3/4} π^{-2} n^{1/4} \exp{π\sqrt{n/6}} ] with the ratios of left- to right-hand side tending to 1 as $n \to \infty$.

math.GR↗
arXiv2004

Spectral computations on lamplighter groups and Diestel-Leader graphs

The Diestel-Leader 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 of the cyclic group of order q with the infinite cyclic group) with respect to a natural generating set. For the "Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and Dicks & Schick have determined the spectrum and the (on-diagonal) spectral measure (Plancherel measure). Here, we show that thanks to the geometric realization, these results can be obtained for all DL-graphs by directly computing an l^2-complete 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 N-step 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 Foelner approximations; in the specific case of DL(q,r), the answer is positive only when r=q.

math.GR↗
arXiv2003

Lie Algebras and Growth in Branch Groups

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 "parabolic space", namely the quotient of the group by the stabilizer of an infinite ray. The growth of the group is bounded from below by the growth of its graded group ring, which connects to the growth of the Lie algebra by a product-sum formula, and the growth of the parabolic space is bounded from below by the growth of the Lie algebra. Finally we use this information to explicitly describe the normal subgroups of the "Grigorchuk group". All normal subgroups are characteristic, and the number of normal subgroups of index $2^n$ is odd and is asymptotically $n^{\log_2(3)}$.

math.GR↗
arXiv2003

Amenability of groups acting on trees

This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation \[Δ= < b,t|[b,t^2]b^{-1},[[[b,t^{-1}],b],b]>.\] I show that it acts transitively on a 3-regular tree, and that $Γ=< b,b^{t^{-1}}$ stabilizes a vertex and acts by restriction on a binary rooted tree. $Γ$ is a "fractal group", generated by a 3-state automaton, and is a discrete analogue of the monodromy action of iterates of f(z)=z^2-1 on associated coverings of the Riemann sphere. $Δ$ shares many properties with the Thompson group $F$. The proof of the main result (amenability of $Δ$) is incomplete in the present form; please refer to the paper arxiv.org/math.GR/0305262, joint with Balint Virag, for a complete proof.

math.GR↗
arXiv2003

Amenability via random walks

We answer an open question of Grigorchuk and Zuk about amenability using random walks. 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.

math.GR↗
arXiv2002

A Wilson Group of Non-Uniformly Exponential Growth

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.

math.GR↗
arXiv2002

From fractal groups to fractal sets

This paper is a survey, with few proofs, of ideas and notions related to self-similarity of groups, semi-groups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, self-similar sets, and renormalizable dynamical systems. In particular, it presents a plausible definition of what a "fractal group" should be, and gives many examples of such groups. A particularly interesting class of examples, derived from monodromy groups of iterated branch coverings, or equivalently from Galois groups of iterated polynomials, is presented. This class contains interesting groups from an algebraic point of view (just-non-solvable groups, groups of intermediate growth, branch groups,...), and at the same time the geometry of the group is apparent in that a limit of the group identifies naturally with the Julia set of the covering map. In its survey, the paper discusses finite-state transducers, growth of groups and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical systems, Hecke-type operators, C^*-algebras, random matrices, ergodic theorems and entropy of non-commuting transformations. Self-similar groups appear then as a natural weaving thread through these seemingly different topics.

math.GR↗
arXiv2002

Cactus Trees, and Estimations of the Spectral Radius of Vertex-Transitive Graphs

This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of ``prime cycles'' at a vertex. The bounds are obtained by constructing circuits in the graph that resemble ``cactus trees'', and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence and underestimation of its spectral radius. The bounds obtained are very good for the Cayley graph of surface groups of genus g>=2, with standard generators (these graphs are the 1-skeletons of tessellations of hyperbolic plane by 4g-gons, 4g per vertex). We have for example for g=2 0.662420<=|M|<=0.662816, and for g=3 0.552773<=|M|<=0.552792.

math.GR↗
arXiv2002

L-presentations and branch groups

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.

math.GR↗
arXiv2002

On a group associated to $z^2-1$

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 $f$.

math.GR↗
arXiv2001

The 2-dimension series of the just-nonsolvable BSV group

I compute the structure of the restricted 2-algebra associated to a group first described by Andrew Brunner, Said Sidki and Ana Cristina Vieira, acting on the binary rooted tree. I show that its width is unbounded, growing logarithmically, and obeys a simple rule. As a consequence, the dimension of the successive quotients of powers of the augmentation ideal in the modular group algebra is p(0)+...+p(n), the count of partitions of numbers up to n.

math.GR↗
arXiv2001

Growth Series and Random Walks on Some Hyperbolic Graphs

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.

math.GR↗
arXiv2001

On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups

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 in irreducible components of the finite quasi-regular representations is given by the double cosets of the parabolic subgroup. Since our results derive from considerations on finite-index subgroups, they also hold for the profinite completions $\hat G$ of the groups G. The representations involved have interesting spectral properties investigated in math.GR/9910102. This paper serves as a group-theoretic counterpart to the studies in the mentionned paper. We study more carefully a few examples of fractal groups, and in doing so exhibit the first example of a torsion-free branch just-infinite group. We also produce a new example of branch just-infinite group of intermediate growth, and provide for it an L-type presentation by generators and relators.

math.GR↗
arXiv2000

Lie Methods in Growth of Groups and Groups of Finite Width

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 examples of groups of finite width and describe their Lie algebras, introducing a notion of Cayley graph for graded Lie algebras. We compute explicitly their lower central and dimensional series, and outline a general method applicable to some other groups from the class of branch groups. These examples produce counterexamples to a conjecture on the structure of just-infinite groups of finite width.

math.GR↗