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Laurent Bartholdi

Publications and source records attributed to Laurent Bartholdi.

At least 19 recordsLinked to original sources
arXiv2026

Wreath products of cocommutative Hopf algebras

We define wreath products of cocommutative Hopf algebras, and show that they enjoy a universal property for coalgebra-split 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. The enveloping algebra of the wreath product of two Lie algebras is the irreducible component containing the identity in the wreath product of their enveloping algebras. We recover the aforementioned result that group extensions may be classified as certain subgroups of a wreath product, and that Lie algebra extensions may also be classified as certain subalgebras of a wreath product.

math.RA↗
arXiv2026

Amenability of Lie, Group and Hopf algebras

We propose to define amenability of a Lie algebra by the existence of almost-invariant finite-dimensional subcoalgebras of its universal enveloping algebra. More generally, a module coalgebra of a cocommutative Hopf algebra is amenable if it admits almost-invariant finite-dimensional subcoalgebras. We prove a coalgebraic rounding theorem: almost-invariant finite-dimensional subspaces can be replaced, without loss in the Følner constant, by almost-invariant finite-dimensional subcoalgebras. For Lie algebras this implies that our definition is equivalent to Elek's amenability of the left regular module of the universal enveloping algebra seen merely as an associative algebra. For groups this recovers the result that a group is amenable if and only if its group ring is algebraically amenable. It furthermore shows that, for every amenable group, all its nonzero modules are amenable, thus proving an assertion by Gromov. We prove that amenable Hopf algebras are closed under taking subalgebras, quotients, cleft extensions, and directed unions, and that every Hopf algebra locally of subexponential growth is amenable. We give examples of amenable Lie algebras which are not elementarily amenable. Finally, we show that amenability passes to the associated graded Hopf-module coalgebra.

math.RA↗
arXiv2026

Super-Arrhenius relaxation of the triangular plaquette model in any dimension

Consider the following plaquette model from statistical physics: a lamp lies at every vertex of the triangular lattice and a switch lies at every even vertex of the (bipartite) dual hexagonal lattice. Each switch toggles the three lamps on its face. The energy of a configuration is the number of ON lamps. For the Glauber dynamics associated with the Gibbs measure defined by this Hamiltonian at any inverse temperature $β>0$, we show that, in any dimension $d\ge 2$, the infinite volume relaxation time satisfies \[e^{β^2/C}/C \le T_{\mathrm{rel}}\le Ce^{e^{Cβ}}\] for some $C>0$. Our result entails that the Gibbs measure is unique. The $e^{β^2}$ scaling was conjectured by Newman and Moore in 1999 and matches the behaviour of supercritical rooted kinetically constrained models such as the East model, thus recovering fragile glass phenomenology in the absence of kinetic constraints. More precisely, we show that, on a torus of side length $2^k$, when $β\to\infty$ and $k/β\to0$, we have $T_{\mathrm{rel}}=e^{2βk(1+o(1))}$. Quite surprisingly, however, we also prove that, on non-periodic finite domains of size $n\le e^{β/C}$ for large $C>0$, we have the much larger asymptotics $\ln T_{\mathrm{rel}}=βn^{Θ(1)}$. The main ingredients of the proofs are new results in extremal and enumerative combinatorics and rely on renormalisation ideas for the dynamics and its groundstates also known as the Ledrappier subshift. We note consequences of our results to geometric group theory (more precisely to the complexity of the word problem for the Baumslag finitely presented group) and to ergodic theory.

math.PR↗
arXiv2026

Automatic actions I. Bounded automata and orbits

We develop the theory of "automatic actions": (semi)groups acting by $ω$-regular transformations on an $ω$-regular language, showing that it covers a large class of heretofore-unrelated examples. We focus on the subclass of actions by "bounded" $ω$-regular transformations, those for which the Büchi automata encoding the action do not have two connected non-trivial cycles. We show that, for bounded actions of inverse semigroups, the orbit relation is also $ω$-regular. We deduce a number of corollaries, in particular decidability, for such actions, of minimality, topological transitivity, aperiodicity, and order of elements. More generally, every first-order statement over the space of the action, involving the action of specific semigroup elements as well as the relation "being in the same orbit", is decidable. We also apply this result to the study of Julia sets of post-critically finite polynomials, and show that the encoding of Fatou components is also computable; thus every first-order statement involving intersection, disjointness etc. of Fatou components or their full orbit under the polynomial, is decidable.

math.GR↗
arXiv2025

Correspondences on Riemann surfaces and non-uniform hyperbolicity

We consider certain correspondences on a Riemann surface, and show that they admit a weak form of hyperbolicity: sufficiently long loops get shorter under lifting at a fixed point and closing. In terms of their algebraic encoding by bisets, this translates to contraction of fundamental group elements along sequences arising from iterated lifting. As an application, we show that apart from the usual Lattès counterexamples, for any rational map on $\mathbb P^1$ with $4$ post-critical points, there is a finite invariant collection of isotopy classes of curves into which every curve is attracted under iterated lifting. More generally, among graphs of given complexity, there exists a finite invariant collect ion of isotopy classes of graphs into which every graph is attracted. Applied to sufficiently rich graphs, the graph attr actor provides a finite set of topological normal forms for the rational map. We also present a strategy towards proving the same statements for maps with more than $4$ post-critical points.

math.DS↗
arXiv2024

An algorithm for uniform generation of unlabeled trees (Pólya trees), with an extension of Cayley's formula

Pólya trees are rooted, unlabeled trees on $n$ vertices. This paper gives an efficient, new way to generate Pólya trees. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with `reality'. Along the way, we give a product formula for the number of rooted labeled trees preserved by a given automorphism; this refines Cayley's formula.

math.CO↗
arXiv2024

Equations in wreath products

We survey solvability of equations in wreath products of groups, and prove that the quadratic diophantine problem is solvable in wreath products of Abelian groups. We consider the related question of determining commutator width, and prove that the quadratic diophantine problem is also solvable in Baumslag's finitely presented metabelian group. This text is a short version of an extensive article by the first-named authors.

math.GR↗
arXiv2024

Snakes can be fooled into thinking they live in a tree

We construct a finitely generated group which is not virtually free, yet has decidable snake tiling problem. This shows that either a long-standing conjecture by Ballier and Stein (the characterization of groups with decidable domino problem as those virtually free ones) is false, or a question by Aubrun and Bitar has a positive answer (there exists a group for which the domino and snake problems are of different difficulty).

math.GR↗
arXiv2024

Groups with presentations in EDT0L

To any family of languages LAN, let us associate the class, denoted $π(\text{LAN})$, of finitely generated groups that admit a group presentation whose set of relators forms a language in LAN. We show that the class of L-presented groups, as introduced by the first author in 2003, is exactly the class of groups that admit presentations in the family of languages EDT0L. We show that the marked isomorphism problem is not semi-decidable for groups given by EDT0L presentations, contrary to the finite presentation case. We then extend and unify results of the first author with Eick and Hartung about nilpotent and finite quotients, by showing that it is possible to compute the marked hyperbolic and marked metabelian quotients of a group given by an EDT0L presentation. Finally, we show how the results about quotient computations enable the construction of recursively presented groups that do not have EDT0L presentations, thus proving $π(\text{EDT0L})\ne π(\text{REC})$. This is done by building a residually nilpotent group with solvable word problem whose sequence of maximal nilpotent quotients is non-computable.

math.GR↗
arXiv2024

Shifts on the lamplighter group

We prove that the lamplighter group admits strongly aperiodic SFTs, has undecidable tiling problem, and the entropies of its SFTs are exactly the upper semicomputable nonnegative real numbers, and some other results. These results follow from two relatively general simulation theorems, which show that for a large class of effective subshifts on the sea-level subgroup, their induction to the lamplighter group is sofic; and the pullback of every effective Cantor system on the integers admits an SFT cover. We exhibit a concrete strongly aperiodic set with $1488$ tetrahedra. We show that metabelian Baumslag-Solitar groups are intersimulable with lamplighter groups, and thus we obtain the same characterization for their entropies.

math.DS↗
arXiv2023

The Topology of Poker

We examine the complexity of the ``Texas Hold'em'' variant of poker from a topological perspective. We show that there exists a natural simplicial complex governing the multi-way winning probabilities between various hands, and that this simplicial complex contains $4$-dimensional spheres as induced subcomplexes. We deduce that evaluating the strength of a pair of cards in Texas Hold'em is an intricate problem, and that even the notion of who is bluffing against whom is ill-defined in some situations.

math.AT↗
arXiv2023

Property (T) and Many Quotients

We prove that, for the free algebra over a sufficiently rich operad, a large subgroup of its group of tame automorphisms has Kazhdan's property (T). We deduce that there exists a group with property (T) that maps onto large powers of alternating groups.

math.GR↗
arXiv2023

The domino problem for hyperbolic groups

We prove, for every non-virtually free hyperbolic group $G$, that there is no algorithm that, given a finite collection of dominoes, determines whether the Cayley graph of $G$ may be edge-covered by these dominoes so that colours match at vertices. This answers a conjecture by Aubrun, Barbieri and Moutot and goes towards settling a long-standing conjecture of Ballier and Stein.

math.GR↗
arXiv2022

On Gardam's and Murray's units in group rings

We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.

math.RA↗
arXiv2022

Growth of groups with linear Schreier graphs

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.

math.GR↗
arXiv2022

Tree languages and branched groups

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.

math.GR↗
arXiv2022

Representation zeta functions of self-similar branched groups

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.

math.GR↗