On representation zeta functions of groups and a conjecture of Larsen and 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 A_2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

7 pages. Research announcement; A shortened version of this version has appeared as N. Avni et al., "On representation zeta functions of groups and a conjecture of Larsen-Lubotzky", C. R. Acad. Sci. Paris, Ser. I (2010), doi:10.1016/j.crma.2010.02.019

References in corpus (2)

• Representation zeta functions of compact p-adic analytic groups and arithmetic groups
• Arithmetic Groups Have Rational Representation Growth