Representation growth of linear groups

Abstract

Let $\Gamma$ be a group and $r_n(\Gamma )$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function \( \calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s} \). When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then \( \calz_\Gamma(s) \) has an ``Euler factorization". The ``factor at infinity" is sometimes called the ``Witten zeta function" counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it is bounded away from $0$. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.