Sixteenth Barcelona Weekend in Group Theory
May 4th to 6th, 2023
The Sixteenth Barcelona Weekend in Group Theory will take place between May 4th (starting in the afternoon) and May 6th (ending before lunch) at the Facultat de Matemàtiques i Estadística of the Universitat Politècnica de Catalunya.
There is no registration fee, but if you are interested in attending, please let us know using the form below.
Confirmed speakers are Laurent Bartholdi (Universität des Saarlandes), María Cumplido (Universidad de Sevilla), Volker Diekert (Universität Stuttgart), Andrew Duncan (Newcastle University), Murray Elder (University of Technology Sydney), Richard Mandel (University of the Basque Country), Ashot Minasyan (University of Southampton), Ignasi Mundet (Universitat de Barcelona), Igor Potapov (University of Liverpool), Francesco Russo (University of Cape Town), Nora Szakács (University of Manchester), and Enric Ventura (Universitat Politècnica de Catalunya).
Schedule
Titles and abstracts
Thursday, May 4th (room 103)
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15:00 - 15:50 Laurent Bartholdi (Saarland University)
The domino problem on groups and Schreier graphs
Consider a directed graph with edge labels in a finite set 𝑆. The domino problem asks, given a finite set of "dots" 𝐴 and a set of "dominos" 𝜃 ⊆ 𝐴 × 𝑆 × 𝐴, whether dots can be assigned to the graph's vertices in such a manner that every edge carries an allowed domino, namely (dots at begin, edge label, dots at end) ∈ 𝜃 for every edge.
This problem is a small fragment of the monadic second order logic of the graph, and we are interested in knowing when it is decidable. In particular, the graph could be the Cayley graph of a group 𝐺 with generating set 𝑆, and then a conjecture of Baller and Stein asserts that the domino problem is decidable if and only if 𝐺 is virtually free. I will describe recent results (partly joint with Ville Salo) proving partial cases of this conjecture.
I will also consider Schreier graphs of 𝐺-sets for some self-similar groups 𝐺, and give examples close to the border between decidability and undecidability of the domino problem.
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16:00 - 16:50 Francesco Russo (University of Cape Town)
Results of Reduction for Commutativity Degrees and Exterior Degrees
The probability of commuting elements $d(G)$, or ``commutativity degree'' of $G$, was introduced by P. Erdős and P. Turan for a finite group $G$ long time ago and indicates how far the group is from being abelian. In fact $d(G)=1$ if and only if $G$ is decomposable in direct product of finitely many cyclic groups of prime power order. The commutativity degree of infinite compact groups was introduced some years later by W. Gustafson and it allows us to detect compact abelian groups; even in this situation structural decompositions appear for the extremal case of probability equal to one.
More recently, P. Niroomand and R. Rezaei introduced the ``exterior degree'' $\widehat{d}(G)$ of $G$, involving the notion of nonabelian exterior square $G \wedge G$ due to R. Brown and J.-L. Loday. It turns out that $\widehat{d}(G)$ is the probability of commuting elements with respect to the operator $\wedge$ of nonabelian exterior square and this produces a notion of commutativity which is weaker than the usual notion of commutativity between two elements of a group. In fact we have $ \widehat{d}(G) \le d(G)$ and the case $\widehat{d}(G)=1$ characterizes $G$ to be a cyclic group. Therefore we are motivated to investigate the exterior degree and its relations with the commutativity degree for infinite groups. In the present talk, I will illustrate some theorems of reduction for the computation of the commutativity degree of infinite groups and the corresponding version for the exterior degree.
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17:00 - 17:50 Maria Cumplido (Universidad de Sevilla)
Rewritings for the word problem in Artin groups
Artin groups are the groups defined by a finite set of generators and relations of the form sts...=tst... where s and t are generators and both words of the equality have the same length. Despite these groups are easily defined, they are quite mysterious: basic problems of classic group theory remain open, as it is the case for the word problem. There have been many (geometric and algebraic) approaches to solve the word problem for particular families of Artin groups (being the braid group the flagship example). In this talk we will explain a method of rewriting words that allows us to obtain geodesic representatives for elements in Artin groups that do not have a relation of length 3 (also known as braid relations) and, as a direct consequence, we will solve the word problem in this (big) family of Artin groups. This is a joint work with Rubén Blasco-García and Rose Morris-Wright.
Friday, May 5th (room 103)
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10:10 - 11:00 Murray Elder (University of Technology Sydney)
Multiple context free languages
Multiple context free languages are a generalisation of content free languages. I will explain what they are, some connections to group theory (which groups have multiple context free word problems, and multiple context free multiplication tables), and give a new ``pumping lemma'' result for showing a language is not multiple context free. This is joint with Andrew Duncan (Newcastle) and Mengfan Lyu (UTS).
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11:10 - 12:00 Richard Evans Mandel (University of the Basque Country)
The quadratic Diophantine problem in certain Baumslag-Solitar groups
The quadratic Diophantine problem in a group G is the problem to decide whether a given quadratic equation has a solution in
G. We show that this problem is NP-complete for (almost all of) the metabelian and unimodular Baumslag-Solitar groups (i.e. BS(1,n) and
BS(n,\pm n)). Additionally, we show that for certain cases, this problem can be solved in polynomial time if the number of variables is
bounded.
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12:10 - 13:00 Ashot Minasyan (University of Southampton)
Product separability in relatively hyperbolic groups
A subset S of a group G is said to be separable if it is closed in the profinite topology. This means that for every element g ∈ G\ S, there exists a finite group M and a homomorphism ϕ:G → M such that ϕ(g) ∉ ϕ(S).
Separability of double cosets of the form QR (where Q,R are subgroups of G) and of more general products is important in many applications. I will mention a few of them in my talk. Afterwards, I will discuss recent work with Lawk Mineh, in which we prove the separability of products of the form Q₁⋯Qₙ, where Q₁,⋯,Qₙ are relatively quasiconvex subgroups in a 'sufficiently nice' relatively hyperbolic group.
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15:00 - 15:50 Igor Potapov (University of Liverpool)
On the Relationship Between Matrix Semigroups, Equations, Maps and Linear Recurrence Sequences
A large number of naturally-defined matrix problems are still unanswered, despite the long history of matrix theory. Some of these questions have recently drawn renewed interest in the context of the analysis of digital processes, verification problems, and links with several fundamental questions in mathematics. In this presentation, I will discuss a number of challenging computational problems for matrix semigroups and their connections to matrix equations, non-deterministic maps and linear recurrence sequences.
For example, let us consider the following problem: Given four n x n matrices A,B,C,D over a ring F (e.g. integer, rational, algebraic numbers), do there exist nonnegative integers x,y,z,w such that A^x B^y C^z D^w= Z, where Z is the zero matrix. In the case where two of these four matrices are identity matrices, the problem is decidable in polynomial time. For a case where only one matrix is identity, the problem is Turing-equivalent to Skolem's problem on reachability in linear recurrent sequences, which is still open in the general case. If none of these matrices are identity matrices the solution of this equation can be non-semilinear and the only decidability result is known for a case of upper triangular 2x2 case. On the other hand, it is known that the problem with a sufficiently large number of matrices of a large dimension is undecidable.
Other matrix semigroup problems create a landscape of fundamental open questions which have natural symbolic, algebraic or geometric interpretations.
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16:00 - 16:50 Nora Szakács (University of Manchester)
Inverse semigroups as metric spaces, and their uniform Roe algebras
Since Gromov's work in the 90s, studying groups as metric spaces has become a main trend in group theory. In turns out that there is a strong link between some of the algebraic properties of the group sand some of the large scale geometric properties of the metric associated to it, and in some cases, this links to properties of a C*-algebra called the uniform Roe algebra associated to the metric space. In the past few years, some of these results have been extended to so-called inverse semigroups by Chung, Gray, Martínez, Lledo, Silva, and the speaker. Inverse semigroups are a generalization of groups which encapsulate partial symmetries similarly as to how groups encapsulate symmetries. During the talk, we will give an introductory exposition of these results.
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17:00 - 17:50 Enric Ventura (Universitat Politècnica de Catalunya)
On the existence of finitely presented intersection-saturated groups
For two subgroups of a group, H₁, H₂ ≤ G, we can look at the eight possibilities for the finite/non-finite generability of H₁, H₂, and H₁ ∩ H₂. For example, all eight are possible in a free non-abelian group except one of them, expressing the well-known fact that free groups are Howson: intersection of two finitely generated subgroups is again finitely generated. A group G is called intersection-saturated when, for every k, each of the 2^(2^k-1) such k-configurations is realizable by appropriate subgroups H₁,...,Hₖ ≤ G.
In this talk, we prove the existence of explicit finitely presented intersection-saturated groups. We also show that the Howson property is the only restriction for realizability in free groups: a k-configuration is realizable in a free non-abelian group if and only if it respects the Howson property.
If time permits, I will explain some ideas to dualize the situation and be able to realize quotient k-configurations (this is still work in progress by the same co-authors J. Delgado and M. Roy).
Saturday, May 6th (room 103)
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10:10 - 11:00 Volker Diekert (Universität Stuttgart)
About the EDT0L-description of solution sets for word equations in graph products
A graph product over groups and monoids generalizes the notion of right-angled Artin group (RAAG) or right-angled Coxeter group (RACG). A fruitful research theme is to consider any property Φ which holds in RAAGs and try to find a natural class 𝒦 of finitely generated monoids for which a transfer result of the following type holds: if Mᵢ, i ∈ J is a finite list of monoids in 𝒦 and if Φ holds for the Mᵢ, then Φ holds for any graph product over the Mᵢ.
If Φ is the property that the existential theory of equations is decidable, then Diekert and Lohrey exhibited a rather large class 𝒦 which included all finitely generated Dedekind-finite monoids (in particular all finitely generated groups) such that the desired result holds (IJAC, Vol. 18, 2008). Here, we are more ambitious. We consider a property Φ which says that we have an effective procedure which represents all solutions of a system of equations and inequalities, for example, as an EDT0L-relation. Then essentially the same 𝒦 as for the existential theory allows the transfer result for EDT0L again.
The talk is based on joint work with Murray Elder (UTS, Sydney) and Markus Lohrey (Uni Siegen).
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11:10 - 12:00 Andrew Duncan (Newcastle University)
Limit groups and Lyndon completions over coherent right-angled Artin groups
Among the main ingredients of Sela's and of Kharlampovich & Miasnikov's proofs of Tarki's problems on the elementary theory of free groups are:
- limit groups over a finitely generated group, which are those that satisfy the same universal sentences as the group; and
- the free ℤ[t]-group F^ℤ[t], or Lyndon completion, of the free group F, which acts as a universe for limit groups over F. (Extending the exponential action of ℤ on a group G to an action by the ring ℤ[t] results in a ℤ[t]-group.)
The strategy is to show the Lyndon completion is a limit group over F and that F embeds into it. The first part of this is achieved by iterating a process of "extending centralisers". The second part uses actions on real trees, in Sela's work, and Makanin-Razborov diagrams in Kharlampovich and Miasnikov's.
This talk will outline how this strategy may be generalized to coherent right-angled Artin groups, describe some of the obstacles along the way, and list some of the consequences. (Joint work with Monste Casals-Ruiz and Ilya Kazachkov.)
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12:10 - 13:00 Ignasi Mundet (Universitat de Barcelona)
Discrete degree of symmetry of manifolds
Let X be a compact connected topological manifold. Let D(X) be the biggest integer m such that X supports an effective continuous action of (Z/r)^m for arbitrarily large values of r. We call D(X) the discrete degree of symmetry of X. Let T(X) be the biggest integer m such that X supports an effective continuous action of the m-torus. It is well known that T(X) is not bigger than dim X, and that equality only holds for tori. T(X) is a lower bound for D(X), but, as we will see, in general D(X) is bigger than T(X). Nevertheless, we wonder whether D(X) satisfies the same inequality as A(X), namely whether D(X) is not bigger than dim X and whether equality only holds for tori. We will explain partial results supporting this expectation.
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