# Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups

Duong H. Dung, Christopher Voll

10 被引用数 (Scopus)

## 抄録

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.

本文言語 英語 6327-6349 23 Transactions of the American Mathematical Society 369 9 https://doi.org/10.1090/tran/6879 出版済み - 2017

• 数学 (全般)
• 応用数学

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