On representation zeta functions of groups and a conjecture of Larsen–Lubotzky

Abstract

We study zeta functions enumerating finite-dimensional irreducible complex linear representations
of compact p-adic analytic and of arithmetic groups. Using methods from p-adic
integration, we show that the zeta functions associated to certain p-adic analytic pro-p
groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding
the abscissa of convergence of arithmetic groups of type A2 defined over number fields,
assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

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