Let be a group and the number of its -dimensional irreducible complex representations. We define and study the associated representation zeta function . When is an arithmetic group satisfying the congruence subgroup property then 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 of the associated simple group over the associated local field . Here we show a surprising dichotomy: if is compact (i.e. anisotropic over ) the abscissa of convergence goes to 0 when goes to infinity, but for isotropic groups it is bounded away from . 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.
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Alexander Lubotzky, Michael Larsen, Representation growth of linear groups. J. Eur. Math. Soc. 10 (2008), no. 2, pp. 351–390DOI 10.4171/JEMS/113