  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.
