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- W4302087642 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 ${sinmathbb{C} mid textrm{Re}(s) = a(G)}$, 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 $mathcal{O}$." @default.
- W4302087642 created "2022-10-06" @default.
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- W4302087642 date "2015-03-24" @default.
- W4302087642 modified "2023-10-16" @default.
- W4302087642 title "Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups" @default.
- W4302087642 doi "https://doi.org/10.48550/arxiv.1503.06947" @default.
- W4302087642 hasPublicationYear "2015" @default.
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