URegina Topology and Geometry Seminar: Francis Bischoff

A standard way of presenting a group is to choose a collection of generators and then to specify a list of relations that must be satisfied. These are not always optimal: sometimes there are non-trivial identities among the relations. For example, the Hall-Witt identity is an identity between the commutators in a free group and gives the non-abelian analogue of the Jacobi identity.

In this talk, I will go over different ways of representing these identities: algebraic, pictorial, topological, and homological. We will then take a look at a number of examples in different groups. In particular, I will cover the case of the Steinberg group, which is related to last week’s talk on K-theory.

The identities among relations are only the first step of an infinite process. Indeed, producing a complete set of identities gives rise to further identities, and this process may continue ad infinitum. Topologically, this corresponds to a cellular decomposition of the classifying space of a group. In several examples, these higher syzygies are organized by a recursive sequence of polytopes, such as cubes, simplices, associahedra, and permutahedra.