Abstract: For a finite group G, let Irr(G) be the set of all irreducible complex characters of G and acd(G) denote the average of the irreducible character degrees of G. It has been proved that if N is a normal subgroup of G and N = A_5, then acd(N) ≤ acd(G). Also, we recently proved that if N = PSL(2,7), then acd(N) ≤ acd(G). In this talk, we conjecture that if N = S^k, where S is a nonabelian simple group, k is a positive integer, and S^k is the direct product of k copies of S, then acd(N) ≤ acd(G). As an evidence, we prove this conjecture for S = A_5 and k = 2. This conjecture, even in the case that k = 1, is unsolved for arbitrary nonabelian simple group. We think that the approach to this problem is unfortunately case-by-case. Also, we study acd(A_n) ≤ acd(S_n), where n ≥ 5. We should mention that if S is a nonabelian simple group such that |Aut(S) : S| is 1 or a prime number, and if we know acd(S) ≤ acd(Aut(S)), then we prove the conjecture for S and k = 1.
Abstract: Given a group G and an automorphism φ of G, two elements x, y ∈ G are said to be φ−conjugate if y = gxφ(g)^{-1} for some g ∈ G. The number of equivalence classes for this relation is the Reidemeister number R(φ) of φ and the set {R(φ) | φ ∈ Aut(G)} is called the Reidemeister spectrum of G. We investigate the Reidemeister spectrum of certain classes of finite groups and analyze the properties a finite group can have when it admits an automorphism with a given Reidemeister number.
Abstract: In the last few years, the model theory of finitely generated groups has experienced a noticeable boost, thanks in part to techniques coming from geometric group theory. In this talk, I’ll present some recent model-theoretic results on selected families of Artin groups, most notably the spherical and affine ones. The focus will be on questions of superstability, as well as on elementary equivalence and first-order rigidity. This is based on joint work with Gianluca Paolini and Giovanni Paolini.
Abstract: Recall that a sequence u in an abelian group G is called a T-sequence (resp., TB-sequence) if there is a Hausdorff (resp., precompact) group topology τ on G such that u → 0 in (G, τ ). Given two T-sequences (resp., TB-sequences) u and v in an abelian group G, we study when their interleaving sequence u |_| v and their sum u + v are T-sequences (resp., T B-sequences) as well. When G = Z and u and v are b-bounded arithmetic sequences, we prove that always u |_| v is a T-sequence and we characterize when it is a TB-sequence. As a consequence, we find a new family of group topologies on Z that are minimally almost periodic. In the particular case when u = (a^n) and v = (b^n) for some integers a, b, we give a complete description of when u |_| v and u + v are T- or TB-sequences.
Abstract: Reversibility, also called Reality, is the property of a group element being conjugate to its inverse. This notion arises naturally in various mathematical and physical contexts, from the time-reversibility of dynamical flows to symmetries in algebraic structures. In this talk, I will survey recent developments in the study of reversibility and its variants. The survey will be based on my recent work in this direction, conducted jointly with collaborators.
Abstract: A category C is said to have the amalgamation property (AP) when each span of monomorphisms in C admits an amalgam. For instance, the categories of groups satisfies the condition AP, while the category of associative algebras over a field does not. The aim of this talk is to investigate the condition AP within the setting of semi-abelian categories, with a particular focus on the category of groups and on varieties of non-associative algebras over a field. We show that the condition AP is closely connected to the representability of actions in C. As an application, we prove that the variety of commutative associative algebras is weakly action representable, while the categories of k-nilpotent (k ≥ 3) and n-solvable (n ≥ 2) groups are not. This is joint work with Jose Brox (Universidad de Valladolid), Xabier García Martínez (Universidade de Santiago de Compostela), Tim Van der Linden (Université catholique de Louvain) and Corentin Vienne (Università degli Studi di Milano Statale).
Abstract: This talk concerns the classification of finite groups in which the centralisers of certain non-central elements are required to be soluble. More precisely, we obtain a complete structural description of finite groups whose non-central element centralisers are all soluble, together with a reduction theorem for groups in which every non-central π-element has a soluble centraliser, for an appropriate set of primes π. These results yield further consequences under mild local assumptions and lead to applications to problems concerning centralisers of involutions and non-commuting graphs of finite groups. This is joint work with Valentina Grazian and Gareth Tracey.
Abstract: Many classical, non-abelian cohomology theories, like the Eilenberg-Mac Lane cohomology of groups or the Hochschild cohomology of associative algebras and Lie algebras, have been unified and understood at a categorical level by means of the so-called direction functor. The use of such functor allows to show that all these cohomology theories are canonically induced by the abelian group structure of their coefficients. In the past decades some monoid cohomology theories have been considered. In such cohomologies, the coefficients are just commutative monoids, and not abelian groups. These cohomologies have a finer classifying power than the classical group-based cohomologies, for instance when they are applied to the classification of topological spaces. The aim of this talk is to show that an adaptation of the direction functor allows to give a categorical, canonical interpretation of the monoid cohomologies, and to describe them by means of suitable monoid extensions, the so-called Schreier extensions.
Abstract: In this talk we address the structure of ambivalent groups, that is, groups in which every element is conjugate to its inverse. After outlining the general features that characterize this class, we present new restrictions that arise in the nilpotent and periodic settings, with particular emphasis on the case where inversion acts fixed-point-freely. We then describe an explicit presentation for ambivalent groups that are not 2-nilpotent, giving a concrete description of their generators and relations. Finally, we discuss how the existence of a strongly embedded subgroup influences the global structure of an ambivalent group.
Abstract: In 1996, Zelmanov conjectured that a just infinite pro-p group of lower central width is either soluble, p-adic analytic, or commensurable to a positive part of a loop group or to the Nottingham group. This conjecture was settled in the negative by Rozhkov, who proved that the (first) Grigorchuk group (and as a consequence also its profinite completion) has lower central width 3. The result by Rozhkov, and later works by Bartholdi and Grigorchuk, do not only determine the lower central width of the Grigorchuk group, but also provide a detailed description of the terms of its lower central series. In this talk, we first survey some known results about the lower central series and lower central width in known groups acting on regular rooted trees such as the aforementioned Grigorchuk group and the Gupta-Sidki 3-group (which is an example of a GGS-group). Then, we present some new results regarding the lower central series and lower central width of a wide class of non-torsion GGS-groups. This is joint work with M. E. Garciarena-Perez and G. A. Fernández-Alcober.
Abstract: Profinite groups can be endowed with a probability measure, which allows one to investigate probabilistic questions in this class of topological groups. A particularly interesting instance is that of a group with a probabilistic identity, that is, when a group satisfies a word with positive probability. In this presentation, I will talk about the structural implications of satisfying a probabilistic identity in two classes of groups: analytic groups and fundamental groups of graphs of pro-p groups. This is joint work with Steffen Kionke, Tommaso Toti, Matteo Vannacci and Thomas Weigel.
Abstract: The interplay between abstract machines and groups has attracted significant attention over the past decades. Two notable examples of this connection are the class of automatic groups and the rational group defined by asynchronous transducers. In this talk, we will explore potential links between them. Particularly interesting examples of automatic groups include hyperbolic groups and CAT(0) cubulated groups, and we will offer further commentary on these special cases.
Abstract: Fusion systems are categories that, in a way, generalize the p-local structure of finite groups and have been proven useful in both algebra and topology. On the other hand Mackey functors are pairs of a covariant and contravariant functor that, in a sense, mimic the transfer, restriction and conjugation maps in group representation theory. Although originally developed to study finite group representation theory, Mackey functors appear in many areas of mathematics such as algebraic k-theory and homotopy theory. Most relevant to this talk is the fact that the cohomology functor can be seen as the contravariant part of a Mackey functor. Following on previous results of Jackowsky and McClure, Díaz and Park conjectured in 2015 that the higher limits of the contravariant part of any Mackey functor (as for example the cohomology functor) over a fusion system vanish. If proven true, this conjecture would solve as a particular case one of the open problems (concerning vanishing of higher limits of the cohomology functor) listed in the 2011 book "Fusion systems in algebra and topology" by Asbacher, Kessar and Oliver. Moreover, a positive result might provide an easier proof of the stable elements theorem as well as provide us with a deeper understanding of fusion systems and, in particular, of exotic fusion systems (i.e. fusion systems which are not realized by a finite group). A negative result to this conjecture would also be of interest as it would provide us with a method of identifying exotic fusion systems without using the classification of finite simple groups. In this talk we present a novel method for approaching this conjecture. More precisely, instead of relying on a result by Jackowsky and McClure, we use the fact that many fusion systems arise from amalgams of finite groups in order to provide a potential inductive method to this conjecture.
Abstract: Let G be a finite group, and let Irr(G) denote the set of the irreducible complex characters of G. An element g of G is called a vanishing element of G if there exists χ∈Irr(G) such that χ(g)= 0 (i.e., g is a zero of χ) and, in this case, the conjugacy class g^G of g in G is called a vanishing conjugacy class. In this talk we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group G such that every non-linear χ∈Irr(G) vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero. Joint work with M.L. Lewis, L. Morotti, E. Pacifici and H.P. Tong-Viet.
Abstract: Let G be a finite group with a generating set S. The diameter of the corresponding Cayley graph is the minimum integer d such that every element of G can be written as the product of at most d elements in S. Let diam(G) be the maximum diameter among all possible generating sets. In this talk we present some new diameter bounds for solvable groups, which imply that transitive solvable groups of degree n have diameter asymptotically less than n^{O(1)}. Interesting examples are given by congruence quotients of the Grigorchuk group. This is all joint work with S. Eberhard, E. Maini and G. Tracey.
Abstract: In this talk, we present our recent findings of a novel three-dimensional chaotic system constructed using only two smooth nonlinear functions. Multi-scroll attractors and a single equilibrium point are among the complicated dynamics that the system displays regardless of its simple construction. With a periodically forcing term, the proposed system reveals a new manifestation of megastability, where an infinite countable family of nested braided attractors emerges. These attractors are topologically analyzed using braid group theory, linking matrices, and symbolic dynamics.
Abstract: The arithmetical properties of the conjugacy class sizes of a finite group, as well as those of the degrees of its irreducible characters, often reveal significant aspects of its algebraic structure. Interestingly, for reasons that remain unclear, these two numerical invariants share several analogies, and such analogies frequently give rise to interesting problems when one attempts to transfer results from one setting to the other. In this talk, an example of this phenomenon will be illustrated; more concretely, the aim is to survey results concerning groups that possess a small number of composite character degrees or class sizes. Particular emphasis will be placed on recent progress in the setting of class sizes, based on joint work with C. Monetta (Università di Salerno).
Abstract: The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. Bruno Duchesne and I examined the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. In particular, we identified them with some kaleidoscopic group acting on dendrites or universal groups acting on biregular trees, realizing them as Polish permutation groups. From these identifications, we deduced algebraic, topological and geometric properties of these groups. This talk will present the identification of the groups and the results of this work.
Abstract: In this talk we discuss a directed graph related to a group G, which we call the N-prime graph Γ_N(G) of G and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of Γ_N(G) are the primes p such that G has an element of order p, and, for distinct vertices p and q, the arc q →p is in the graph if and only if G has a subgroup of order p whose normalizer in G has an element of order q. Generalizing some known results about the Gruenberg-Kegel graph, we will see that the group V(ZG) of the units with augmentation 1 in the integral group ring ZG has the same N-prime graph as G if G is a finite solvable group, and we reduce to almost simple groups the problem of whether Γ_N(V(ZG)) = Γ_N(G) holds for any finite group G. We also prove that Γ_N(V(ZG)) = Γ_N(G) if G is almost simple with socle either an alternating group, or PSL(2,r^f) with r prime and f ≤2. Finally, for G solvable we obtain some stronger results which give a contribution to the Subgroup Isomorphism Problem. More precisely, we prove that if V(ZG) contains a Frobenius subgroup T with kernel of prime order and complement of prime-power order, then G contains a subgroup isomorphic to T. Joint work with E. Pacifici and A. del Rio.
Abstract: For a finitely generated group G and a finite symmetric generating system Σ with length function ℓ its growth function γ_{G,Σ}(t), given by the sum of t^{ℓ(g)} for g∈G, is a formal power series which has been the subject of intensive investigations by A. Mann and I. Chiswell. By van Danztig’s theorem every totally disconnected locally compact (=t.d.l.c.) group G contains a compact open subgroup O. Let R denote a set of double coset representatives of O\G/O and let µ denote a left invariant Haar measure of G. For r∈R call ℓ(r) = µ(OrO)/µ(O) the length of the double coset OrO. One says that G has bounded double coset growth if a_k = |{r ∈R | ℓ(r) = k}| is finite for all natural numbers k. This property is independent of the choosen left invariant Haar measure µ as well as of the choosen compact open subgroup O of G. In the talk we discuss some recent results and open problems concerning the formal Dirichlet series ζ_{G,O}(s) (the sum of a_n(n^s) for all natural numbers n) for a t.d.l.c. group of bounded double coset growth. Joint work with George Willis, Ilaria Castellano and Gianmarco Chinello.
Abstract: Finitely axiomatizing a group G within a class of (profinite) groups C consists of isolating G inside C, up to (continuous) isomorphism, by means of a single first-order sentence in the language of groups L_gp, or in a suitable expansion thereof. In a class of pro-p groups endowed with a manifold structure over an appropriate pro-p domain, this naturally leads to the question of which geometric or analytic invariants are first-order definable. In earlier work, Conte and Klopsch showed that, within the class of p-adic analytic pro-p groups, the manifold dimension is first-order definable: more precisely, for each integer d ≥ 0 there exists a first-order sentence φ_d in L_gp such that a p-adic analytic pro-p group G satisfies φ_d if and only if dim(G) = d. In this talk we discuss analogous phenomena for analytic groups defined over more general pro-p domains. We present positive results under the additional hypothesis that the underlying coefficient ring has characteristic zero, or that the language of groups has been suitably extended. Finally, we highlight obstructions that arise in positive characteristic and indicate several directions for further investigation. This is joint work with Martina Conte.