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

TL;DR

This paper proves that the representation zeta functions of finitely generated torsion-free nilpotent groups have uniform analytic properties, including rational abscissa of convergence and meromorphic continuation, with implications for understanding their representation growth.

Contribution

It establishes uniform analytic properties of representation zeta functions for a broad class of finitely generated nilpotent groups, showing invariance of key analytic invariants.

Findings

Zeta functions have rational abscissa of convergence.

Meromorphic continuation to the left of the abscissa.

Invariance of convergence abscissa and pole order across groups.

Abstract

Let $G$ be a finitely generated torsion-free 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 $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$, the continued function is holomorphic except for a pole at $s=a(G)$. 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 more general result establishing uniform analytic properties of representation zeta functions of finitely generated nilpotent groups of the form $\mathbf{G}(\mathcal{O})$, where $\mathbf{G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ 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 $\mathbf{G}$, independent of $\mathcal{O}$.