Department of Mathematics

The geometry of spaces of representations

Exploring the properties of groups using matrices and their geometry.

Group theory is the abstract study of symmetry. One way to understand an arbitrary group is to embed it in a group of matrices, since matrices are concrete objects which are easy to compute with. A representation of a group H is a homomorphism from H to G, where G is an algebraic matrix group or a Lie group. Representations often arise when H acts on a geometric object such as a manifold, an algebraic variety or a combinatorial building.

It turns out it is fruitful to study not just individual representations, but the space of all representations. In many cases, this space has a rich geometric structure. We study the geometric and arithmetic properties of spaces of representations. There are applications to the subgroup structure of algebraic groups, geometric invariant theory, representation growth of discrete groups and lattices in automorphism groups of trees. This research draws on ideas from many areas of mathematics, including algebraic geometry (the geometry of polynomials), differential geometry, group theory, graph theory, number theory and model theory (a branch of mathematical logic).

Researchers at the University of Auckland

This research is supported by the New Zealand Marsden Fund.