Advanced Seminar on Group Theory

Lehrstuhl für Algebra und Zahlentheorie

Summer semester 2024

Where & When:
The usual time is Tuesday 10:30 - 12:00 in seminar room 03.73.
Content:
Advanced topics in and around group theory are discussed in short series of talks and guest lectures.
Organizers:
Prof. Dr. Benjamin Klopsch & Martina Conte & Margherita Piccolo

Tue 16.04.24
Seminar room 03.73
all of us
Discussion of topics for this semester
Tue 23.04.24
Seminar room 03.73
Clotilde Gauthier (Besançon)
Classification of the irreducible representations of  GL(2,q) 

The aim of this talk is to build every irreducible representations of GL(2,q) on a field with characteristic prime to q. In order to do that, we first study the representations induced by linear characters of the Borel subgroup and then, we extend the (q-1)-dimensional representation of the mirabolic subgroup.

Analytic Group Theory

In this seminar, we delve into different analytic approaches to study groups and their actions. Our journey will be guided by the book [1], authored by Cohen and Gelander, which serves as our roadmap. According to the authors, this book is an introductory text to the topic, but it can also be used "as a lightweight book to take for a one month trek in the Nepali Himalayas".

We will mostly focus our attention on amenability, one of the most important properties of analytic group theory. In the last part of the seminar we will introduce property (T). If we have time, we will also look at the Tits alternative.

See the programme for more details.

Main reference:

  1. T. Cohen and T. Gelander, An Invitation to Analytic Group Theory, 2024. link
Tue 30.04.24
Seminar room 03.73
Lucrezia Prosperi
Amenability
Tue 07.05.24
Seminar room 03.73
Nadja Hempel
Nets
Tue 14.05.24
Seminar room 03.73
Martina Conte
Actions of Amenable Groups
Tue 21.05.24
Seminar room 03.73
Doris Grothusmann
Growth of groups
Tue 28.05.24
Seminar room 03.73
Mikel Eguzki Garciarena (Bilbao)
Lower Central Series of the GGS-groups

Groups of automorphisms of regular rooted trees are a rich source of examples with interesting properties in group theory, and they have been used to solve very important problems. The first Grigorchuk group, defined by Grigorchuk in 1980, is one of the first instances of an infinite finitely generated periodic group, thus providing a negative solution to the General Burnside Problem. It is also the first example of a group with intermediate growth, hence solving the Milnor Problem. Many other groups of automorphisms of rooted trees have since been defined and studied. Important examples are the Gupta-Sidki p-groups, for p an odd prime, and the second Grigorchuk group. These are again finitely generated infinite periodic groups and they belong to the large family of the so-called Grigorchuk-Gupta-Sidki groups (GGS-groups, for short).

Some research has been done regarding the lower central series of groups of automorphisms of regular rooted trees and more specifically of some particular GGS-groups, but the knowledge is scarce. The aim of this talk is to present some of the techniques and tools that we have developed to understand the lower central series of some of the GGS-groups that act on the p-adic tree, and how do we use them to completely determine the lower central series of some GGS-groups such us the p-Fabrykowski-Gupta groups.

This is a joint work with Gustavo Fernández-Alcober and Marialaura Noce.

Tue 04.06.24
Seminar room 03.73
TBA
Amenability of topological groups
Tue 11.06.24
Seminar room 03.73
Hongyi Huang (Bristol)
Permutations, bases and low rank groups

Let G = GL(V), where V is a finite-dimensional vector space, and recall that any element in G is uniquely determined by its action on a basis for V. In addition, any two pairs of linearly independent vectors can be mapped to each other by an element of G. These two basic linear algebra properties can be interpreted in the language of permutation groups, which leads us naturally to the definitions of base and rank of a permutation group. In this talk, I will present some of my recent results on bases for primitive permutation groups, and I will report on recent progress with C.H. Li and Y.Z. Zhu towards a classification of the rank three groups.

Tue 18.06.24
Seminar room 03.73
Alena Meyer
Prefrattini subgroups and Crowns

TBA

Tue 25.06.24
Seminar room 03.73
Olga Varghese
Property (T)
Tue 02.07.24
Seminar room 03.73
Anitha Thillaisundaram
(Lund)
Invariable generation of certain branch groups

A group is invariably generated if there is some generating set S such that upon replacing any number of elements of S with arbitrary conjugates of these elements, it still results in a generating set for the group. All finite groups are invariably generated, but this is not the case for infinite groups. For example, Kantor, Lubotzky and Shalev showed that a finitely generated linear group is invariably generated if and only if it is virtually solvable. In this talk, we investigate invariable generation among some classical examples of branch groups, such as the Grigorchuk-Gupta-Sidki groups; this is joint work with Charles Cox. Recall that branch groups are well-studied groups acting on rooted trees, that first arose as 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.

Tue 09.07.24
Seminar room 03.73
TBA
(Optional) The Tits alternative
Tue 16.07.24
Seminar room 03.73
Ettore Marmo
(Milano Bicocca)
Hyperplane arrangements and Bloch-Kato pro-p groups

For every field K we can define a distinguished extension Ksep called its separable closure. The maximal pro-p quotient GK(p) of the Galois group GK = Gal(Ksep/K) is called the maximal pro-p Galois group of K, many arithmetical properties of the field are encoded in the structure of this group. It is interesting to ask which pro-p groups can be realized as the maximal pro-p Galois group of some field. It is known that any such group must also satisfy a cohomological property called the Bloch-Kato property.

In this talk we will discuss some families of pro-p groups arising from hyperplane and toric arrangements and some techniques to study the Bloch-Kato property in this context. This is based on joint work with Th. Weigel and E. Delucchi.

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