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.

# Past Junior Topology and Group Theory Seminar

I will talk about the boundaries of CAT(0) groups giving definitions, some examples and will state some theorems. I may even prove something if there is time.

I will discuss a couple of techniques often useful to prove quasi-isometric rigidity results for isometry groups. I will then sketch how these were used by B. Kleiner and B. Leeb to obtain quasi-isometric rigidity for the class of fundamental groups of closed locally symmetric spaces of noncompact type.

We will discuss various familiar properties of groups studied in geometric group theory, whether or not they are invariant under quasi-isometry, and why.

One can ask whether the fundamental groups of 3-manifolds are distinguished by their sets of finite quotients. I will discuss the recent solution of this question for Seifert fibre spaces.

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.

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.

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.

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.

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.