TY - JOUR

T1 - Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups

AU - Dung, Duong H.

AU - Voll, Christopher

N1 - Funding Information:
The authors acknowledge support from the DFG Sonderforschungsbereich 701 at Bielefeld University. They thank Sebastian Herr, Benjamin Martin and Tobias Rossmann for several helpful discussions and an anonymous referee for very valuable feedback. The second author thanks the University of Auckland and the Alexander von Humboldt Foundation for support during the final phase of work on this paper.
Publisher Copyright:
© 2017 American Mathematical Society.

PY - 2017

Y1 - 2017

N2 - Let G be a finitely generated nilpotent group. The representation zeta function ζG(s) of G enumerates twist isoclasses of finite-dimensional irreducible complex representations of G. We prove that ζG(s) has rational abscissa of convergence α(G) and may be meromorphically continued to the left of α(G) and that, on the line {s ∈ ℂ | Re(s) = α(G)}, the continued function is holomorphic except for a pole at s = α(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.

AB - Let G be a finitely generated nilpotent group. The representation zeta function ζG(s) of G enumerates twist isoclasses of finite-dimensional irreducible complex representations of G. We prove that ζG(s) has rational abscissa of convergence α(G) and may be meromorphically continued to the left of α(G) and that, on the line {s ∈ ℂ | Re(s) = α(G)}, the continued function is holomorphic except for a pole at s = α(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.

UR - http://www.scopus.com/inward/record.url?scp=85020446194&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020446194&partnerID=8YFLogxK

U2 - 10.1090/tran/6879

DO - 10.1090/tran/6879

M3 - Article

AN - SCOPUS:85020446194

SN - 0002-9947

VL - 369

SP - 6327

EP - 6349

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 9

ER -