Past Junior Topology and Group Theory Seminar

2 March 2016
16:00
Alex Margolis
Abstract

I will present a basic overview of finiteness conditions, group cohomology, and related quasi-isometry invariance results. In particular, I will show that if a group satisfies certain finiteness conditions, group cohomology with group ring coefficients encodes some structure of the `homology at infinity' of a group. This is seen for hyperbolic groups in the work of Bestvina-Mess, which relates the group cohomology to the Čech cohomology of the boundary.

  • Junior Topology and Group Theory Seminar
20 January 2016
16:00
Federico Vigolo
Abstract

I will illustrate how to build families of expanders out of 'very mixing' actions on measure spaces. I will then define the warped cones and show how these metric spaces are strictly related with those expanders.

  • Junior Topology and Group Theory Seminar
2 December 2015
16:00
Nicolaus Heuer
Abstract

Quasihomomorphisms (QHMs) are maps $f$ between groups such that the
homomorphic condition is boundedly satisfied. The case of QHMs with
abelian target is well studied and is useful for computing the second
bounded cohomology of groups. The case of target non-abelian has,
however, not been studied a lot.

We will see a technique for classifying QHMs $f: G \rightarrow H$ by Fujiwara and
Kapovich. We will give examples (sometimes with proofs!) for QHM in
various cases such as

  • the image $H$  hyperbolic groups,
  • the image $H$ discrete rank one isometries,
  • the preimage $G$ cyclic / free group, etc.

Furthermore, we point out a relation between QHM and extensions by short
exact sequences.

  • Junior Topology and Group Theory Seminar
25 November 2015
16:00
Federico Vigolo
Abstract

A family of expanders is a sequence of finite graphs which are both sparse and highly connected. Firstly defined in the 80s, they had huge applications in applied maths and computer science. Moreover, it soon turned out that they also had deep implications in pure maths. In this talk I will introduce the expander graphs and I will illustrate a way to construct them by approximating actions of groups on probability spaces.

  • Junior Topology and Group Theory Seminar
18 November 2015
16:00
Kieran Calvert
Abstract

In 1964 Golod and Shafarevich discovered a powerful tool that gives a criteria for when a certain presentation defines an infinite dimensional algebra. In my talk I will assume the main machinery of the Golod-Shafarevich inequality for graded algebras and use it to provide counter examples to certain analogues of the Burnside problem in infinite dimensional algebras and infinite groups. Then, time dependent, I will define the Tarski number for groups relating to the Banach-Tarski paradox and show that we can using the G-S inequality show that the set of Tarski numbers is unbounded. Despite the fact we can only find groups of Tarski number 4, 5 and 6.

  • Junior Topology and Group Theory Seminar
11 November 2015
16:00
Robert Kropholler
Abstract

I will discuss a notoriously hard problem in group theory known as the flat closing conjecture. This states that a group with a finite classifying space is either hyperbolic or contains a Baumslag-Solitar Subgroup. I will give some strategies to try and create a counterexample to this conjecture. 

  • Junior Topology and Group Theory Seminar

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