Representation Growth of Groups.

For a group $G$, let $a_n(G)$ be the number of n-dimensional
representations of $G$. For certain kinds of groups ($SL(n,Z)$ for example), the
behavior of $a_n(G)$ can be encoded in a zeta-like function. I will describe
several results and open questions regarding the poles (which give the rate
of growth of $a_n(G)$), meromorphic continuation (which gives the error
terms), and functional equation (which give surprising symmetries between
the $a_n(G)$’s) of this zeta function.