Arithmetic groups

Let G be a group and let r(n,G) denote the number of equivalence classes of n-dimensional complex irreducible representations of G. Representation growth is a branch of asymptotic group theory that studies the asymptotic and arithmetic properties of the sequences (r(n,G)). In 2008 Larsen and Lubotzky conjectured that all irreducible lattices in a high rank semisimple Lie group have the same degree of polynomial representation growth. In this talk I will explain the conjecture and describe the ideas around the proof of a variant of the conjecture: if the lattices have polynomial representation growth (which is known to be true in most cases) then they have the same degree of polynomial growth. This is a joint work with Nir Avni, Benjamin Klopsch and Christopher Voll.