Past Junior Topology and Group Theory Seminar

Starting from a definition of the cohomology of a group, we will define the bounded cohomology of a group. We will then show how quasi-homomorphisms lead to cocycles in the second bounded cohomology group, and use this to look at the second bounded cohomology of some of our favourite groups. If time permits we will end with some applications.

I will introduce Symmetric spaces via a result of Kleiner & Leeb, comparing the axioms in their definition of a Euclidean building with properties of symmetric spaces of noncompact type.

Take a group G and split it as the fundamental group of a graph of groups, then take the vertex groups and split them as fundamental groups of graphs of groups etc. If at some point you end up with a collection of unsplittable groups, then you have a hierarchy. Haken showed that for any 3-manifold M with an incompressible surface S, one can cut M along S and and then find other incompressible surfaces in M\S and cut again, and repeating this process one eventually obtains a collection of balls. Analogously, Delzant and Potyagailo showed that for any finitely presented group without 2-torsion and a certain sensible class E of subgroups of G, G admits a hierarchy where the edge groups of the splittings lie in E. I really like their proof and I will present it.

The Alexander polynomial of a link was the first link polynomial. We give some ways of defining this much-studied invariant, and derive some of its properties.

We introduce Outer space, a contractible finite dimensional topological space on which the outer automorphism group of a free group acts 'nicely.' We will explain what 'nicely' is, and provide motivation with comparisons to symmetric spaces, analogous spaces associated to linear groups.

We will look at CAT(0) spaces, their isometries and boundaries (defined through Busemann functions).