Tag - Group theory

Henry Bradford: Quantitative LEF and topological full groups

9th November, 2020

Topological full groups of minimal subshifts are an important source of exotic examples in geometric group theory, as well as being powerful invariants of symbolic dynamical systems. In 2011, Grigorchuk and Medynets proved that TFGs are LEF, that is, every finite subset of the multiplication table occurs in the multiplication table of some finite group. In this talk we explore some ways in which asymptotic properties of the finite groups which occur reflect asymptotic properties of the associated subshift.

Hung Nguyen: Group characters and their fields of values

9th November, 2020

Representations and characters have been proved to be very useful tools to study finite groups. I will discuss some results on the relationship between fields of values of characters of a finite group and the structure of the group, from a classical result of Burnside on real-valued characters and Navarro-Tiep's result on rational-valued characters of odd degree to very recent results of my collaborators and myself on (almost) p-rational characters.

Alan Logan: Free groups via graphs

5th November, 2020

An LMS online lecture course in free groups and graph theory.

Free groups may be viewed as the fundamental groups of graphs. This observation allows for a very intuitive view of free groups and their subgroups. These lectures combine topological ideas, due to Stallings in the 1980s, with more combinatorial and computational ones to prove many of the fundamental results in free groups. These results include the Nielsen-Schreier Theorem (subgroups of free groups are free), Howson's Theorem (finitely generated subgroups have finitely generated intersection), and the decidability of the subgroup membership problem.

Olga Kharlampovich: Fraïssé limits of limit groups

29th October, 2020

We modify the notion of a Fraïssé class and show that various interesting classes of groups, notably the class of non-abelian limit groups and the class of finitely generated elementary free groups, admit Fraïssé limits. We will also discuss countable elementary free groups.

Tasho Kaletha: An explicit supercuspidal local Langlands correspondence

29th October, 2020

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any p-adic group G that splits over a tame extension, provided p does not divide the order of the Weyl group. This construction matches any discrete Langlands parameters with trivial monodromy to an L-packet consisting of supercuspidal representations, and describes the internal structure of these L-packets.

The construction has two parts. The depth-zero part involves generalizing to disconnected groups results of Lusztig on the decomposition of a non-singular Deligne-Lusztig induction. Higher multiplicities occur in this decomposition and are handled using work of Bonnafé-Dat-Rouquier. The positive-depth part involves functorial transfer from a twisted Levi subgroup, which is made possible by an improvement of Yu's construction of supercuspidal representations obtained in recent joint work with Fintzen and Spice, and consideration of Harish Chandra characters.

We will also discuss ongoing work towards related conjectures: Shahidi's generic L-packet conjecture, Hiraga-Ichino-Ikeda formal degree conjecture,  stability and endoscopic transfer.

Daniele Garzoni: Invariable generation of finite simple groups

25th October, 2020

A subset X of a group G invariably generates G if we are free to replace each element of X by an arbitrary conjugate, and we must always obtain a generating set of G. This concept was introduced by Dixon in the early nineties with motivations from computational Galois theory. We will review these motivations and their intimate connections with permutation groups. We will then present some new results concerning the probability of generating invariably a finite simple group. For instance, we will see that two random elements of a finite simple group of Lie type of bounded rank invariably generate with probability bounded away from zero.

Marialaura Noce: Groups of automorphisms of rooted trees

22nd October, 2020

An LMS online lecture course in groups acting on trees.

Groups of automorphisms of rooted trees have been studied for years as an important source of groups with interesting properties. For instance, the Grigorchuk group (that is a group acting on the binary tree) is the first example of a finitely generated group with intermediate growth (this answered an open question posed by Milnor) and the first example of an amenable but not elementary amenable group. Furthermore, this group provides a counterexample to the General Burnside Problem.

In these lectures we will first introduce the basic theory of groups of automorphisms of rooted trees and their subgroups. Then we will give examples and main properties of such groups, including the aforementioned Grigorchuk group, and the GGS groups.