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

Journal Article


Abstract


  • 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.

Publication Date


  • 2017

Geographic Focus


Citation


  • Dung, D. H. & Voll, C. (2017). Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups. Transactions of the American Mathematical Society, 369 (9), 6327-6349.

Scopus Eid


  • 2-s2.0-85020446194

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1949

Number Of Pages


  • 22

Start Page


  • 6327

End Page


  • 6349

Volume


  • 369

Issue


  • 9

Place Of Publication


  • United States

Abstract


  • 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.

Publication Date


  • 2017

Geographic Focus


Citation


  • Dung, D. H. & Voll, C. (2017). Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups. Transactions of the American Mathematical Society, 369 (9), 6327-6349.

Scopus Eid


  • 2-s2.0-85020446194

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1949

Number Of Pages


  • 22

Start Page


  • 6327

End Page


  • 6349

Volume


  • 369

Issue


  • 9

Place Of Publication


  • United States