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


Authors: Duong H. Dung and Christopher Voll
Journal: Trans. Amer. Math. Soc. 369 (2017), 6327-6349
MSC (2010): Primary 20F18, 20E18, 22E55, 20F69, 11M41
DOI: https://doi.org/10.1090/tran/6879
Published electronically: March 1, 2017
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Abstract: 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\in \mathbb{C} \mid \mathrm {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 $ \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}$.


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Additional Information

Duong H. Dung
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Email: dhoang@math.uni-bielefeld.de

Christopher Voll
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Email: voll@math.uni-bielefeld.de

DOI: https://doi.org/10.1090/tran/6879
Keywords: Finitely generated nilpotent groups, representation zeta functions, Kirillov orbit method, $p$-adic integrals
Received by editor(s): May 13, 2015
Received by editor(s) in revised form: September 23, 2015
Published electronically: March 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society