Research lines

The three main lines of research of the group are Group Theory, Topology and its Applications.

Group Theory

The research interests of the group cover different parts of group theory and representation theory. More precisely, we are interested in infinite groups (from geometric, algebraic, combinatorial, model theoretical and asymptotic viewpoint), topological groups (especially profinite groups) and finite groups and their representation.

G1. Geometric and Combinatorial Group Theory

The field of geometric group theory emerged from Gromov’s insight that even mathematical objects such as groups, which are defined completely in algebraic terms, can be profitably viewed as geometric objects and studied with geometric techniques. Contemporary geometric group theory has broadened its scope considerably but retains this basic philosophy of reformulating in geometric terms problems from diverse areas of mathematics and then solving them with a variety of tools. The growing list of areas where this general approach has been successful includes low-dimensional topology, the theory of manifolds, algebraic topology, complex dynamics, combinatorial group theory, algebra, logic, the study of various classical families of groups, Riemannian geometry and representation theory.

In this area, our main interests are:

- Right-angled Artin groups and their generalizations.

Right-angled Artin groups (RAAGs for short) have become central in group theory and are one of the most representative exponents of the interplay between geometric group theory and other areas of mathematics. This class interpolates between two of the most classical families of groups, free and free abelian groups, and its study provides uniform approaches and proofs, as well as rich generalisations of the results for free and free abelian groups. The study of this class from the different perspectives has lead to the development of new modern theories such as the theory of CAT(0) cube complexes -which has been an essential ingredient in Agol’s solution to the virtually fibred Conjecture and the recent theory of hierarchically hyperbolic groups -a vast generalization of the class of hyperbolic groups that, among others, contains the families of mapping class groups. In this line of research, we study RAAGs from geometric, algebraic, model theoretical and asymptotic points of view.

- Groups acting on rooted trees: GGS groups and related families.

The subgroups of the group of automorphism of a d-adic tree are an important source of groups with interesting properties. For example, it is easy to produce subgroups which are finitely generated, periodic, and infinite, thus giving a negative answer to the General Burnside Problem. More precisely, one can give examples which are p-groups, for p a prime.

The first such examples are due to Grigorchuk for p=2 and to Gupta and Sidki for p>2. They have been generalized to a parameterized family of groups, the so-called GGS-groups, which act on the d-adic tree and depend on a vector

e=(e₁,…,e_d-1) with values in Z/dZ. Not all GGS-groups are periodic, but there is a simple criterion to determine periodicity.

In this line of research, we are interested in the class of GGS-groups and more specifically the characterization of GGS-groups that have the congruence subgroup property and the study of the Engel-condition.

G2. Finite and Profinite groups

- Profinite groups: zeta functions, representation growth and word problems.

Zeta functions.

Asymptotic group theory deals with the growth of some functions counting certain invariants in infinite groups (e.g. finite index subgroups, normal subgroups, representations, …). If the growth of these invariants is polynomial we can define a zeta function associated to them as it is done for the Riemann zeta function. In this line, we are interested in the study of the zeta functions of rational representations compact p-adic analytic groups. In the case of SL(F_p[[t]],n), it was conjectured that the zeta function associated to its

irreducible complex representations coincides with that of SL(Z_p,n).

In group theory, a word can be thought of as an element of the free group in countably many variables. Word problems have gained a new impetus in group theory in the last decade; a milestone is the proof of Ore's conjecture in 2010, almost 60 years after it was formulated.

Representation growth.

In a finite group the number of rational irreducible representations coincides with the number of conjugacy classes of cyclic subgroups. We expect that studying the growth of rational representations of free groups and surface groups will allow us to solve a conjecture posed by Bujalance, Conder, Gamboa, Gromadzki and Izquierdo about the number of surfaces for which a given Riemann surface is a double covering. We expect to extend this conjecture to cyclic coverings.

Word problem.

A classical word problem, dating back to Philip Hall, is to determine whether a given word w is concise in some class C of groups, i.e. whether the following holds for every group G in C: if w only takes finitely many values in G, is the verbal subgroup w(G) finite? Ivanov constructed a non-concise word in the class of all groups, but it is still an open question whether all words are concise in the class of residually finite groups. This major problem is related to the following question: if G is a d-generator finite group and w takes m values in G, is the order of w(G) bounded by a function of d and m, depending on w but not on G? Some progress has been made by Guralnick and Shumyatsky, who showed in 2014 that all words of the form […[x₁ⁿ¹,x₂]ⁿ²,x₃]ⁿ³,…,x_k]^nk are concise in the class of residually finite groups.

Another question is the study of the number N(G,w) of solutions to the equation w=1 in a finite group G. If G is abelian, then N(G,w)≥|G|^k-1, where k is the number of variables in w. Amit conjectured that the same bound holds in all finite nilpotent groups. This was independently confirmed by Levy and Íñiguez for groups of class 2. More generally, if a is a value of w in G, it makes sense to ask whether the size of the fiber consisting of the solutions to the equation w=a is also at least |G|^k-1.

We are also interested in questions about the commutator word w=[x,y]. In 1982, Guralnick proved that, if G is a finite p-group and G’ is abelian, then all elements of G’ are commutators provided that d(G’)≤2, or that d(G’)≤3 and p≥5. There are methods in Guralnick's proof that resemble those of the theory of powerful p-groups, even if at some points it is essential to use the commutativity of G'.

A way to attack the first problem in the previous section could be the use of profinite methods, by embedding a residually finite group in its profinite completion. On the other hand, it is possible to formulate a genuinely profinite version of Hall's conciseness problem as follows: if G is a profinite group in which a word w takes countably many values, is then w(G) finite? This question has been considered by Detomi, Morigi, and Shumyatsky, who have given a positive answer for outer commutator words.

- Finite groups: cohomology and characters of finite p-groups.

Cohomology of finite p-groups

The coclass of a finite p-group is the difference between the exponent of the order of the group and its nilpotency class. Carlson proved that for p=2 and a fixed positive integer c, the number of isomorphism types of rings that appear as mod p cohomology rings of a finite p-group of coclass c is finite. He also conjectured the same fact for odd primes. The main ingredient of this approach is to construct a universal object together with a quasi-isomorphism for the complex associated to abelian p-groups of small rank.

In this direction, if q and p are different prime numbers, by a result of Quillen the isomorphism type of the mod q cohomology rings of the groups SL(Z/p^kZ,n) does not depend on k. A similar behaviour is expected for the case p=q when n is smaller than p.

On the other hand, a classical result of Tate states that a finite group G is p-nilpotent if and only if the restriction map in cohomology from G to its Sylow p-subgroups is an isomorphism in degree one. Our goal is to extend this nilpotency criterion to p-solvability. Natural applications to bounding the p-length of a p-solvable group then appear, extending a result of Huppert to the prime p=2.

Characters of finite p-groups.

There are multiple directions for research on characters of finite p-groups. One such direction is related to the degrees of the irreducible characters, their field of values, the zeroes of characters, etc. In this case the main goal is to study characters themselves. on other occasions, characters are used to prove group theoretical results. In this direction, characters can be used to study word problems (an area which is also considered in the research line G2.3) in p-groups (or in groups in general). A natural way of doing this is by considering the function N_w which, given a group G and a word w, counts the number of solutions of the equations w=g for g in G. The objective is to study to which extent these functions are characters of G and obtain information on N_w and G.

Topology

T1. Pointfree topology

Pointfree Topology is a well-established area of study of what is usually called Categorical Topology. It is a modern algebraic approach to Topology on a constructive foundation, using the tools of Category Theory and Lattice Theory, with nice and interesting ramifications in Logic and Computation (because it can be dealt in an impredicative constructive setting, e.g. in a topos). One of the main uses of Pointfree Topology is to reformulate definitions and results from classical mathematics, replacing the standard point-set notions with the corresponding point-free notions (points and continuous functions are derived notions; instead basic open sets and their covering and approximations relations are taken as fundamental). This idea has been pursued in mathematics for almost 30 years, and is quite common within computer science and topos theory.

T2. Multi-valued and non-commutative topology

Many-valued Topology (another terminology: Lattice-valued Topology or Fuzzy Topology) replaces sets by maps with values in a complete lattice. If we agree that the point of general topology is the existence of important nonmetrizable topologies, then of similar sense is lattice-valued topology: there are important lattice-valued topologies which cannot be represented as two-valued ones. Another new quality appears when the complete lattice also has an algebraic structure with a not necessarily commutative operation (e.g. when it is a quantale). This gives rise to noncommutative lattice-valued topology - a subject which may enter areas of mathematics which are nonaccesible to the traditional commutative two-valued topologies. In particular, there are convergences which are never two-topological but which can be topologized in the framework of a noncommutative lattice-valued topology. Other such areas include nonspatial locales and categorical topology among others.

T3. Numerical representations

The analysis of numerical representations of ordered structures, in particular of interval orders and semiorders has many applications coming from quite different scientific areas as Entropy (Physics), Utility Theory (Economics), Theoretical Computer Science, Artificial Intelligence or Extensive Measurement Theory (Psychology). The theoretical questions to be analysed use many mathematical branches as Order Theory and General Topology, among others.

Applications

A1. Applications to medicine

Discrete mathematics is finding applications in many areas of biology and medicine.

An application that is getting increasing importance in medicine is vaccine design, so that one can find rational methods to obtain vaccines in which the most frequent epitopes are covered and, at the same time, the epitopes are covered in a balanced way. This improves the efficiency of the vaccines, especially in cases where the pathogen has high mutation rates, which is the case, for instance, of HIV virus for which although a big progress in the search of an efficient vaccine has been made, nevertheless no completely effective one has been found yet.

Another application which is receiving interest in biotechnology is the design of DNA-codes with a big number of words and satisfying also that the melting temperature of all the words is similar. These codes have applications in many fields related to bioengineering, like for instance the design of biological bar codes of applications to DNA-computing.

We plan to apply combinatorics and discrete mathematics, using techniques coming from algebra, graph theory and combinatorial optimization, to get applications in the two above mentioned areas: to vaccine design, further developing the study of lambda-superstrings initiated by L. Martínez et al., and also to finding DNA-codes with many words and similar melting temperature for the words in the code.

A2. Applications to Computer Science

Cryptography has appeared with the development of civilization, but it has never been as crucial as it is today. Since the invention and expansion of the internet, every day we transfer huge amounts of private data that requires encryption. Thus we have a strong demand for cryptoanalysis of public key cryptosystems.

Security of cryptographic protocols relies on the “difficulty” of solving certain problems and this naturally brings one to the question of how we measure the complexity of a problem. During the 90’s group theorists and computer scientists working on the Computational Group Theory Package “Magnus” noticed that many problems that are considered to be theoretically difficult, could be attacked computationally using simple strategies. These observations motivated the study of different ways of measuring complexity.

In their work I. Kapovich, A. Miasnikov (external collaborator of our group) and P. Schupp introduced the notion of generic case complexity. The key practical idea of this notion is that a problem must be considered “difficult” if it is difficult on almost all inputs (on a generic set) and we should ignore the computational complexity on sparse sets of atypical inputs. This natural approach proved to be very interesting, since the authors managed to show that a large class of classical algorithmically unsolvable problems, in practice can be solved in linear time on generic sets, thereby showing that many of the proposed cryptosystems are not secure for most inputs. This points to many new interesting questions such as the existence of generically difficult problems as well as questions what the right notion of genericity is.

The group intends to open a new line of research with Distinguished Research Professor and Department director Alexei Miasnikov (Stevens Institute, USA) - a leader in the study of generic case complexity, and Director of Research at CNRS and Director of LaBRI Pascal Weil (Bordeaux, France) - a leader in the study of generic sets and subgroups of free groups.