University of Sydney Algebra Seminar

Uri Onn (Ben Gurion University)

Friday 17 April, 12:05-12:55pm, Place: Carslaw 375

Representation zeta functions of arithmetic groups

Let G be a group and let r(n,G) denote the number of isomorphism classes of n-dimensional complex irreducible representations of G. Representation growth is a branch of asymptotic group theory which studies the asymptotic and arithmetic properties of the sequence (r(n,G)). Whenever this sequence grows polynomially one can associate to it a Dirichlet generating function, known as the representation zeta function of G. Larsen and Lubotzky proved that for arithmetic groups which have polynomial representation growth the associated zeta functions have an Euler product decomposition allowing local-global analysis. One can then apply a variety of tools such as the Kirillov orbit method, p-adic integration, Algebraic geometry, Model theory and Clifford theory. In this talk I will explain how these ingredients fit together to give some interesting properties of representation zeta functions associated to arithmetic and p-adic groups.