Abstract: A group is called representation rigid if it has only finitely many irreducible complex representations of each finite dimension. If the numbers of such representations of a rigid group grow slowly, i.e. polynomially, we may consider them as the coefficients of a Dirichlet generating function, called the group's representation zeta function.

In my talk I will discuss some aspects of the theory of representation zeta functions of arithmetic groups. These zeta functions share a number of features with their classical predecessors in number theory and geometry. Often, for instance, they are Euler products, indexed by the places of a number field, whose non-archimedean factors are rational functions.

As I will explain, the study of these Euler factors and the analytic properties of their products calls for methods which are quite distinct from those used in more classical contexts. I will discuss a range of tools -- ranging from (p-adic) Lie theory, geometric representation theory, to p-adic integration and model theory -- which were developed to study the Euler factors representation zeta functions of arithmetic groups, and their local and global properties.