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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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  • [1] Avraham Aizenbud and Nir Avni, Representation growth and rational singularities of the moduli space of local systems, Invent. Math. 204 (2016), no. 1, 245-316. MR 3480557, https://doi.org/10.1007/s00222-015-0614-8
  • [2] Nir Avni, Arithmetic groups have rational representation growth, Ann. of Math. (2) 174 (2011), no. 2, 1009-1056. MR 2831112, https://doi.org/10.4007/annals.2011.174.2.6
  • [3] Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Representation zeta functions of compact $ p$-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), no. 1, 111-197. MR 3011874, https://doi.org/10.1215/00127094-1959198
  • [4] Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Arithmetic groups, base change, and representation growth, Geom. Funct. Anal. 26 (2016), no. 1, 67-135. MR 3494486, https://doi.org/10.1007/s00039-016-0359-6
  • [5] Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Similarity classes of integral $ \mathfrak{p}$-adic matrices and representation zeta functions of groups of type $ \mathsf {A}_2$, Proc. Lond. Math. Soc. (3) 112 (2016), no. 2, 267-350. MR 3471251, https://doi.org/10.1112/plms/pdv071
  • [6] J. Denef, On the degree of Igusa's local zeta function, Amer. J. Math. 109 (1987), no. 6, 991-1008. MR 919001, https://doi.org/10.2307/2374583
  • [7] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$ p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368
  • [8] Marcus du Sautoy and Fritz Grunewald, Analytic properties of zeta functions and subgroup growth, Ann. of Math. (2) 152 (2000), no. 3, 793-833. MR 1815702, https://doi.org/10.2307/2661355
  • [9] S. Ezzat,
    Representation growth of finitely generated torsion-free nilpotent groups: Methods and examples,
    PhD thesis, University of Canterbury, New Zealand, 2012.
  • [10] Shannon Ezzat, Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field, J. Algebra 397 (2014), 609-624. MR 3119241, https://doi.org/10.1016/j.jalgebra.2013.08.028
  • [11] F. J. Grunewald, D. Segal, and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), no. 1, 185-223. MR 943928, https://doi.org/10.1007/BF01393692
  • [12] E. Hrushovski, B. Martin, and S. Rideau,
    Definable equivalence relations and zeta functions of groups, with an appendix by R. Cluckers.
    Preprint, arXiv:0701011, 2015.
  • [13] A. Jaikin-Zapirain, Zeta function of representations of compact $ p$-adic analytic groups, J. Amer. Math. Soc. 19 (2006), no. 1, 91-118 (electronic). MR 2169043, https://doi.org/10.1090/S0894-0347-05-00501-1
  • [14] Benjamin Klopsch, Representation growth and representation zeta functions of groups, Note Mat. 33 (2013), no. 1, 107-120. MR 3071315
  • [15] Michael Larsen and Alexander Lubotzky, Representation growth of linear groups, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 351-390. MR 2390327, https://doi.org/10.4171/JEMS/113
  • [16] Alexander Lubotzky and Andy R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985), no. 336, xi+117. MR 818915, https://doi.org/10.1090/memo/0336
  • [17] Alexander Lubotzky and Benjamin Martin, Polynomial representation growth and the congruence subgroup problem, Israel J. Math. 144 (2004), 293-316. MR 2121543, https://doi.org/10.1007/BF02916715
  • [18] Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR 1978431
  • [19] Charles Nunley and Andy Magid, Simple representations of the integral Heisenberg group, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp. 89-96. MR 982280, https://doi.org/10.1090/conm/082/982280
  • [20] Tobias Rossmann, Topological representation zeta functions of unipotent groups, J. Algebra 448 (2016), 210-237. MR 3438311, https://doi.org/10.1016/j.jalgebra.2015.09.050
  • [21] Michael M. Schein and Christopher Voll, Normal zeta functions of the Heisenberg groups over number rings II--the non-split case, Israel J. Math. 211 (2016), no. 1, 171-195. MR 3474960, https://doi.org/10.1007/s11856-015-1271-8
  • [22] Michael M. Schein and Christopher Voll, Normal zeta functions of the Heisenberg groups over number rings I: the unramified case, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 19-46. MR 3338607, https://doi.org/10.1112/jlms/jdu061
  • [23] Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786
  • [24] R. Snocken,
    Zeta functions of groups and rings,
    PhD thesis, University of Southampton, 2014.
  • [25] A. Stasinski and C. Voll, Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type $ B$, Amer. J. Math. 136 (2014), no. 2, 501-550. MR 3188068, https://doi.org/10.1353/ajm.2014.0010
  • [26] Christopher Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. (2) 172 (2010), no. 2, 1181-1218. MR 2680489, https://doi.org/10.4007/annals.2010.172.1185
  • [27] Christopher Voll, A newcomer's guide to zeta functions of groups and rings, Lectures on profinite topics in group theory, London Math. Soc. Stud. Texts, vol. 77, Cambridge Univ. Press, Cambridge, 2011, pp. 99-144. MR 2807857
  • [28] Christopher Voll, Zeta functions of groups and rings--recent developments, Groups St Andrews 2013, London Math. Soc. Lecture Note Ser., vol. 422, Cambridge Univ. Press, Cambridge, 2015, pp. 469-492. MR 3495675
  • [29] Don Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math. 83 (1986), no. 2, 285-301. MR 818354, https://doi.org/10.1007/BF01388964

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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

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