arXiv subjects
Publications and source records attributed to Laurent Bartholdi.
We introduce a new method of proving upper estimates of growth of finitely generated groups and constructing groups of intermediate growth using graphs of their actions. These estimates are of the form $\exp(n^α)$ for some $α<1$, and provide the first examples of such bounds for simple groups of intermediate growth.
We prove the connectedness of the following locus: the space of degree-$d$ branched self-coverings of $S^2$ with two critical points of order $d$, one of which is $n$-periodic.
We study the portraits of isometries of rooted trees - the labelling of the tree, at each vertex, by the permutation of its descendants - in terms of languages. We characterize regularly branched self-similar groups in terms of $ω$-regular languages. We deduce the algorithmic decidability of some problems, such as the comparison of regularly branched contracting groups, and their orbit structure on the boundary of the rooted tree.
We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the coëfficients 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 for some prominent examples of branch groups, such as the Grigorchuk and Gupta-Sidki groups.
Let $F$ be a free group. We present for arbitrary $g\in\mathbb{N}$ a LogSpace (and thus polynomial time) algorithm that determines whether a given $w\in F$ is a product of at most $g$ commutators; and more generally an algorithm that determines, given $w\in F$, the minimal $g$ such that $w$ may be written as a product of $g$ commutators (and returns $\infty$ if no such $g$ exists). The algorithm also returns words $x_1,y_1,\dots,x_g,y_g$ such that $w=[x_1,y_1]\cdots[x_g,y_g]$. The algorithms we present are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a conjecture by Bardakov.
We introduce a notion of "simulation" for labelled graphs, in which edges of the simulated graph are realized by regular expressions in the simulating graph, and prove that the tiling problem (aka "domino problem") for the simulating graph is at least as difficult as that for the simulated graph. We apply this to the Cayley graph of the "lamplighter group" $L=\mathbb Z/2\wr\mathbb Z$, and more generally to "Diestel-Leader graphs". We prove that these graphs simulate the plane, and thus deduce that the seeded tiling problem is unsolvable on the group $L$. We note that $L$ does not contain any plane in its Cayley graph, so our undecidability criterion by simulation covers cases not covered by Jeandel's criterion based on translation-like action of a product of finitely generated infinite groups. Our approach to tiling problems is strongly based on categorical constructions in graph theory.
We construct a finitely presented group $G$ such that the $7$th dimension quotient $G\cap(1+\varpi(\mathbb Z G)^7)/γ_7(G)$ has an element of order $3$; this immediately leads to a finite $3$-group without the dimension property. This contradicts a series of results by N. Gupta.
We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the "dimension problem" by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary $s,d$ the torsion of the homotopy group $π_s(S^d)$ into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, since for every prime $p$, there is some $p$-torsion in $π_{2p}(S^2)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group $π_4(S^2)=\mathbb Z/2\mathbb Z$. We finally obtain analogous results in the context of Lie rings: for every prime $p$ there exists a Lie ring with $p$-torsion in some dimension quotient.
We consider the domino problem on Schreier graphs of self-similar groups, and more generally their monadic second-order logic. On the one hand, we prove that if the group is bounded then the graph's monadic second-order logic is decidable. This covers, for example, the Sierpiński gasket graphs and the Schreier graphs of the Basilica group. On the other hand, we already prove undecidability of the domino problem for a class of self-similar groups, answering a question by Barbieri and Sablik, and some examples including one of linear growth.
May's Theorem (1952), a celebrated result in social choice, provides the foundation for majority rule. May's crucial assumption of symmetry, often thought of as a procedural equity requirement, is violated by many choice procedures that grant voters identical roles. We show that a weakening of May's symmetry assumption allows for a far richer set of rules that still treat voters equally. We show that such rules can have minimal winning coalitions comprising a vanishing fraction of the population, but not less than the square root of the population size. Methodologically, we introduce techniques from group theory and illustrate their usefulness for the analysis of social choice questions.
We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their word problem is $\mathsf{NC}^1$-hard. For some of these groups (including Grigorchuk's group and Thompson's groups) we prove that the compressed word problem (which is equivalent to the circuit evaluation problem) is $\mathsf{PSPACE}$-complete.
Let $G$ be the first Grigorchuk group. We show that the commutator width of $G$ is $2$: every element $g\in [G,G]$ is a product of two commutators, and also of six conjugates of $a$. Furthermore, we show that every finitely generated subgroup $H\leq G$ has finite commutator width, which however can be arbitrarily large, and that $G$ contains a subgroup of infinite commutator width. The proofs were assisted by the computer algebra system GAP.
We compute the $p$-central and exponent-$p$ series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to a partially commuting polynomial ring and power series ring. We finally show how the growth series of these various objects are related to each other.
We introduce self-similar algebras and groups closely related to the Thue-Morse sequence, and begin their investigation by describing a character on them, the "spread" character.
Call a group action on a topological space \emph{biminimal} if for any points $x,y\in X$ there exists a group element taking $x$ arbitrarily close to $y$ and whose inverse takes $y$ arbitrarily close to $x$. A symbolic encoding of the Thue-Morse dynamical system is given, in terms of $ω$-automata. It is used to prove that the Thue-Morse dynamical system is minimal but not biminimal. The $ω$-automata also establish a link between Nekrashevych's presentation of limit spaces and solenoids with a construction described by Vershik and Solomyak.
We consider "Thurston maps": branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, 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 problems in a mapping class biset. Our decomposition results on mapping class bisets reduce these decision problems to small mapping class bisets; they correspond to rational maps, homeomorphisms and maps double covered by a torus endomorphism, and their conjugacy and centralizer problems are solvable respectively in terms of complex analysis, group theory and linear algebra. Branched coverings themselves are also encoded into bisets, with actions of the fundamental groups. We characterize those bisets that arise from branched coverings between topological spheres, and extend this correspondence to maps between spheres with multicurves, whose algebraic counterparts are "sphere trees of bisets". To illustrate the difference between Thurston maps and homeomorphisms, we produce a Thurston map with infinitely generated centralizer --- while centralizers of homeomorphisms are always finitely generated.
We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata. We prove that the Order Problem is decidable when the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in the whole family. The former results are implemented and will be available in the GAP package FR developed by the first author.
We address the problem of determining the class of self-similar groups, and in particular its closure under restricted direct products. We show that the group $\mathbb Z^{(ω)}$ is self-similar, that $G^{(ω)}\rtimes C_2$ is self-similar whenever $G$ is, and that permutational wreath products of a finite abelian group with a self-similar group are self-similar.