Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups
Let G be a finitely generated nilpotent group. The representation zeta function zeta(G)(s) of G enumerates twist isoclasses of finite-dimensional irreducible complex representations of G. We prove that zeta(G)(s) has rational abscissa of convergence alpha(G) and may be meromorphically continued to the left of alpha(G) and that, on the line {s is an element of C vertical bar Re(s) = alpha(G)}, the continued function is holomorphic except for a pole at s = alpha(G). A Tauberian theorem yields a precise asymptotic result on the representation growth of G in terms of the position and order of this pole. We obtain these results as a consequence of a result establishing uniform analytic properties of representation zeta functions of torsion-free finitely generated nilpotent groups of the form G(O), where G is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring O of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of G, independent of O.
369
9
6327-6349
6327-6349
Amer Mathematical Soc