A partial linear space (PLS) is a point-line incidence structure such that each line is incident with at least two points and each pair of points is incident with at most one line. We say that a PLS is proper if there exists at least one non-collinear point pair, and at least one line incident with more than two points. The highest degree of symmetry for a proper PLS occurs when the automorphism group G is transitive on ordered pairs of collinear points, and on ordered pairs of non-collinear points. In this case, G is a transitive rank 3 group on the points. While the primitive rank 3 PLSs are essentially classified, we present the first substantial classification of a family of imprimitive rank 3 examples. We classify all imprimitive rank 3 proper partial linear spaces such that the rank 3 group is innately transitive (including quasiprimitive cases) or semiprimitive and induces an almost simple group on the unique nontrivial system of imprimitivity. We construct several infinite families of examples and ten individual examples. The examples admit a rank 3 action of a linear or unitary group, and to our knowledge most of our examples have not appeared before in the literature. This is a joint work with Alice Devillers and Cheryl Praeger.
The notion of strong conciseness of a group-word extends the classical concept of conciseness from abstract groups to the profinite setting. A word is said to be strongly concise in a class of profinite groups if, for any , the cardinality of the set of values taken by in being strictly smaller than implies that the verbal subgroup of is finite. In this talk we will study the relation between this notion and the notion of equationally Noetherian groups. These groups arise from the theory of algebraic geometry over groups, which we will develop throughout the talk. As a consequence, we will see that every word is strongly concise in the class of profinite linear groups, as well as in the class of profinite completions of virtually abelian-by-polycyclic groups. This is joint work with Andoni Zozaya.
I will give a quick introduction into the classical finiteness conditions and for a discrete group and then explain how to extend these to certain topological groups. The search for discrete groups that are of type but not of type has a very interesting and rich history. In this talk will present a new family of discrete and topological groups with this property. This is joint work with I. Castellano, B. Marchionna, and Y. Santos-Rego.
Formal Concept Analysis (FCA) is a branch of applied lattice theory, concerned with the study of concept hierarchies derived from collections of objects and their attributes. Introduced by R. Wille in the 1980s, FCA now has found applications in machine learning and related fields. An application of FCA to hyperplane arrangements yields a new Galois connection on the (conjugacy classes of) parabolic subgroups of a finite reflection group. Combined with methods from Serre’s recent work on involution centralizers, we obtain a refinement of Howlett’s description of the normalizers of parabolic subgroups of a finite Coxeter group. This is joint work G. Roehrle and J.M. Douglass.
The representation growth of a group measures the asymptotic distribution of its irreducible representations. Whenever the growth is polynomial, a suitable vehicle for studying it is a Dirichlet generating series called the representation zeta function of . One of the key invariants in this context is the abscissa of convergence of the representation zeta function. The spectrum of all abscissae arising across a given class of groups is of considerable interest and has been studied in some cases. In the realm of -adic analytic groups (with perfect Lie algebra), the abscissae of convergence are explicitly known only for groups of small dimensions. But there are interesting asymptotic results for “simple” -adic analytic groups of increasing dimension. In this talk, I will give an overview of the main tools and ingredients in this area and I will report on recent work joint with Moritz Petschick to enlarge the class of groups.
The family of multi-EGS groups form a natural generalisation of the Grigorchuk-Gupta-Sidki groups, which in turn are well-studied groups acting on rooted trees. Groups acting on rooted trees provided the first explicit examples of infinite finitely generated torsion groups, and since then have established themselves as important infinite groups, with numerous applications within group theory and beyond. Among these groups with the most interesting properties are the so-called regular branch groups. In this talk we investigate the normal subgroups in non-torsion regular branch multi-EGS groups, and we show that the congruence completion of these multi-EGS groups have bounded finite central width. In particular, we prove that the profinite completion of a Fabrykowski-Gupta group has width 2. This is joint work with Benjamin Klopsch.
Counting the number of subgroups in a finite group has numerous applications, ranging from enumerating certain classes of finite graphs (up to isomorphism), to counting how many isomorphism classes of finite groups there are of a given order. In this talk, I will discuss the history behind the question; why it is important; and what we currently know.