Past Algebra Seminar

4 March 2014
17:00
Dr David Craven
Abstract
The maximal subgroups of the exceptional groups of Lie type have been studied for many years, and have many applications, for example in permutation group theory and in generation of finite groups. In this talk I will survey what is currently known about the maximal subgroups of exceptional groups, and our recent work on this topic. We explore the connection with extending morphisms from finite groups to algebraic groups.
25 February 2014
17:00
Abstract
We define the notion of orbit decidability in a general context, and descend to the case of groups to recognise it into several classical algorithmic problems. Then we shall go into the realm of free groups and shall analise this notion there, where it is related to the Whitehead problem (with many variations). After this, we shall enter the negative side finding interesting subgroups which are orbit undecidable. Finally, we shall prove a theorem connecting orbit decidability with the conjugacy problem for extensions of groups, and will derive several (positive and negative) applications to the conjugacy problem for groups.
18 February 2014
17:00
Chris Parker
Abstract
In this talk, I will explain part of the programme of Gorenstein, Lyons and Solomon (GLS) to provide a new proof of the CFSG. I will focus on the difference between the initial notion of groups of characteristic $2$-type (groups like Lie type groups of characteristic $2$) and the GLS notion of groups of even type. I will then discuss work in progress with Capdeboscq to study groups of even type and small $2$-local odd rank. As a byproduct of the discussion, a picture of the structure of a finite simple group of even type will emerge.
4 February 2014
17:00
Tim Riley
Abstract
For a finitely presented group, the Word Problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is the time-complexity of a direct attack on the Word Problem by applying the defining relations. A "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. I will explain why, nevertheless, there are efficient (polynomial time) solutions to the Word Problems of these groups. The main innovation is a means of computing efficiently with compressed forms of enormous integers. This is joint work with Will Dison and Eduard Einstein.
10 December 2013
17:00
Richard Weidmann
Abstract
We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).
19 November 2013
17:00
Francesco Matucci
Abstract
Given a residually finite group, we analyse a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how ``efficient'' of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this talk I will introduce these growths and give an overview of some cases and properties. This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.

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