Abstract
For the unitary dual mapping of an automorphism of a torsion-free, finite rank nilpotent group, we prove the Pólya–Carlson dichotomy between rationality and the natural boundary for the analytic behavior of its Artin–Mazur dynamical zeta function. We also establish Gauss congruences for the Reidemeister numbers of the iterations of endomorphisms of groups in this class. Our method is the twisted Burnside–Frobenius theorem proven in the paper for automorphisms of this class of groups, and a calculation of the Reidemeister numbers via a product formula and profinite completions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Bell, R. Miles, and Th. Ward, “Towards a Pólya-Carlson Dichotomy for Algebraic Dynamics”, Indag. Math. (N.S.), 25:4 (2014), 652–668.
Jakub Byszewski and Gunther Cornelissen, “Dynamics on Abelian Varieties in Positive Characteristic”, Algebra Number Theory, 12:9 (2018), 2185–2235.
A. Fel’shtyn, E. Troitsky, and M. Zietek, “New Zeta Functions of Reidemeister Type and the Twisted Burnside-Frobenius Theory”, Russ. J. Math. Phys., 27:2 (2020), 199–211.
A. Fel’shtyn, “Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion”, Mem. Amer. Math. Soc., 147:699 (2000).
A. Fel’shtyn and R. Hill, “The Reidemeister Zeta Function with Applications to Nielsen Theory and a Connection with Reidemeister Torsion”, K-Theory, 8:4 (1994), 367–393.
A. Fel’shtyn and B. Klopsch, “Pólya–Carlson Dichotomy for Coincidence Reidemeister Zeta Functions via Profinite Completions”, arXiv:2102.10900, (2021).
A. Fel’shtyn and E. Troitsky, “Twisted Burnside-Frobenius Theory for Discrete Groups”, J. Reine Angew. Math., 613 (2007), 193–210.
A. Fel’shtyn and E. Troitsky, “Twisted Burnside-Frobenius Theory for Endomorphisms of Polycyclic Groups”, Russian J. Math. Phys., 25:1 (2018), 17–26.
A. Fel’shtyn, E. Troitsky, and A. Vershik, “Twisted Burnside Theorem for Type II\({}_1\) Groups: an Example”, Math. Res. Lett., 13:5 (2006), 719–728.
D. Gonçalves, “The Coincidence Reidemeister Classes on Nilmanifolds and Nilpotent Fibrations”, Topology Appl., 83 (1998), 169–186.
D. Gonçalves and P. Wong, “Twisted Conjugacy Classes in Nilpotent Groups”, J. Reine Angew. Math., 633 (2009), 11–27.
N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, vol. 58, Grad. Texts in Math., second edition, Springer-Verlag, New York, 1984.
A. Lubotzky and D. Segal, Subgroup Growth, vol. 212, Birkhäuser Verlag, Basel, 2003.
Derek J. S. Robinson, A Course in the Theory of Groups, vol. 80, Springer-Verlag, New York, 1996.
Sanford L. Segal, Nine Introductions in Complex Analysis, vol. 208, Elsevier Science B.V., Amsterdam, revised edition, 2008.
A. Weil, Basic Number Theory, Grundlehren Math. Wiss., Band 144, Springer-Verlag New York, Inc., New York, 1967.
Funding
The work of Alexander Fel’shtyn is funded by the Narodowe Centrum Nauki of Poland (NCN) (grant no. 2016/23/G/ST1/04280(Beethoven2)). The work of Evgenij Troitsky was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS.”
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Fel’shtyn, A., Troitsky, E. Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted Burnside–Frobenius Theorem. Russ. J. Math. Phys. 28, 455–463 (2021). https://doi.org/10.1134/S1061920821040051
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920821040051