Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion

Alexander Fel'shtyn; Richard Hill

Banach Center Publications (1999)

  • Volume: 49, Issue: 1, page 77-116
  • ISSN: 0137-6934

Abstract

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In this paper we prove trace formulas for the Reidemeister numbers of group endomorphisms and the rationality of the Reidemeister zeta function in the following cases: the group is finitely generated and the endomorphism is eventually commutative; the group is finite; the group is a direct sum of a finite group and a finitely generated free Abelian group; the group is finitely generated, nilpotent and torsion free. We connect the Reidemeister zeta function of an endomorphism of a direct sum of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the unitary dual map, and as a consequence obtain a connection of the Reidemeister zeta function with Reidemeister torsion. We also prove congruences for Reidemeister numbers which are the same as those found by Dold for Lefschetz numbers.

How to cite

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Fel'shtyn, Alexander, and Hill, Richard. "Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion." Banach Center Publications 49.1 (1999): 77-116. <http://eudml.org/doc/208970>.

@article{Felshtyn1999,
abstract = {In this paper we prove trace formulas for the Reidemeister numbers of group endomorphisms and the rationality of the Reidemeister zeta function in the following cases: the group is finitely generated and the endomorphism is eventually commutative; the group is finite; the group is a direct sum of a finite group and a finitely generated free Abelian group; the group is finitely generated, nilpotent and torsion free. We connect the Reidemeister zeta function of an endomorphism of a direct sum of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the unitary dual map, and as a consequence obtain a connection of the Reidemeister zeta function with Reidemeister torsion. We also prove congruences for Reidemeister numbers which are the same as those found by Dold for Lefschetz numbers.},
author = {Fel'shtyn, Alexander, Hill, Richard},
journal = {Banach Center Publications},
keywords = {Reidemeister torsion; trace formula; Reidemeister numbers; Nielsen numbers; zeta function},
language = {eng},
number = {1},
pages = {77-116},
title = {Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion},
url = {http://eudml.org/doc/208970},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Fel'shtyn, Alexander
AU - Hill, Richard
TI - Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion
JO - Banach Center Publications
PY - 1999
VL - 49
IS - 1
SP - 77
EP - 116
AB - In this paper we prove trace formulas for the Reidemeister numbers of group endomorphisms and the rationality of the Reidemeister zeta function in the following cases: the group is finitely generated and the endomorphism is eventually commutative; the group is finite; the group is a direct sum of a finite group and a finitely generated free Abelian group; the group is finitely generated, nilpotent and torsion free. We connect the Reidemeister zeta function of an endomorphism of a direct sum of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the unitary dual map, and as a consequence obtain a connection of the Reidemeister zeta function with Reidemeister torsion. We also prove congruences for Reidemeister numbers which are the same as those found by Dold for Lefschetz numbers.
LA - eng
KW - Reidemeister torsion; trace formula; Reidemeister numbers; Nielsen numbers; zeta function
UR - http://eudml.org/doc/208970
ER -

References

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  1. [1] D. Anosov, The Nielsen number of maps of nilmanifolds, Russian Math. Surveys 40 (1985), 149-150. Zbl0594.55002
  2. [2] M. Artin and B. Mazur, On periodic points, Ann. of Math. 81 (1965), 82-99. 
  3. [3] I. Babenko and S. Bogatyi, Private communication. 
  4. [4] J. S. Birman, Braids, links and mapping class groups, Ann. Math. Studies 82, Princeton Univ. Press, Princeton, 1974. 
  5. [5] R. Bowen and O. Lanford, Zeta functions of restrictions of the shift transformation, Proc. Global Anal. 1968, 43-49. Zbl0211.56501
  6. [6] J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109 (1979), 259-322. Zbl0412.58026
  7. [7] A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419-435. Zbl0583.55001
  8. [8] C. Epstein, The spectral theory of geometrically periodic hyperbolic 3-manifolds, Mem. AMS, 335 (1985). Zbl0584.58047
  9. [9] E. Fadell and S. Husseini, The Nielsen number on surfaces, in: Topological methods in nonlinear functional analysis, Contemp. Math. 21, AMS, 1983, 59-98. Zbl0563.55001
  10. [10] A. Fathi and M. Shub, Some dynamics of pseudo-Anosov diffeomorphisms, Astérisque 66-67 (1979), 181-207. Zbl0446.57022
  11. [11] A. L. Fel'shtyn, New zeta function in dynamics, in: Tenth Internat. Conf. on Nonlinear Oscillations, Varna, Abstracts of Papers, Bulgar. Acad. Sci., 1984, 208. 
  12. [12] A. L. Fel'shtyn, A new zeta function in Nielsen theory and the universal product formula for dynamic zeta functions, Funktsional. Anal. i Prilozhen. 21 (2) (1987), 90-91 (in Russian); English transl.: Functional Anal. Appl. 21 (1987), 168-170. 
  13. [13] A. L. Fel'shtyn, Zeta functions in Nielsen theory, Funktsional. Anal. i Prilozhen. 22 (1) (1988), 87-88 (in Russian); English transl.: Functional Anal. Appl. 22 (1988), 76-77. 
  14. [14] A. L. Fel'shtyn, New zeta functions for dynamical systems and Nielsen fixed point theory, in: Lecture Notes in Math. 1346, Springer, 1988, 33-55. 
  15. [15] A. L. Fel'shtyn, The Reidemeister zeta function and the computation of the Nielsen zeta function, Colloq. Math. 62 (1991), 153-166. 
  16. [16] A. L. Fel'shtyn and R. Hill, Dynamical zeta functions, Nielsen theory and Reidemeister torsion, in: Nielsen Theory and Dynamical Systems, Contemp. Math. 152, AMS, 1993, 43-69. Zbl0793.58028
  17. [17] A. L. Fel'shtyn and R. Hill, The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion, K-theory 8 (1994), 367-393. Zbl0814.58033
  18. [18] A. L. Fel'shtyn, R. Hill and P. Wong, Reidemeister numbers of equivariant maps, Topology Appl. 67 (1995), 119-131. Zbl0845.55002
  19. [19] A. L. Fel'shtyn and V. B. Pilyugina, The Nielsen zeta function, Funktsional. Anal. i Prilozhen. 19 (4) (1985), 61-67 (in Russian); English transl.: Functional Anal. Appl. 19 (1985), 300-305. Zbl0603.58041
  20. [20] J. Franks, Homology and Dynamical Systems, CMBS Regional Conf. Ser. Math. 49 (1982). 
  21. [21] W. Franz, Über die Torsion einer Überdeckung, J. Reine Angew. Math. 173 (1935), 245-254. 
  22. [22] D. Fried, Periodic points and twisted coefficients, in: Lect. Notes in Math. 1007 (1983), 261-293. 
  23. [23] D. Fried, Homological identities for closed orbits, Invent. Math. 71 (1983), 219-246. 
  24. [24] D. Fried, Lefschetz formula for flows, in: The Lefschetz centennial conference, Contemp. Math. 58, AMS, 1987, 19-69. 
  25. [25] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. 53 (1981), 53-78. Zbl0474.20018
  26. [26] M. Gromov, Hyperbolic groups, in: Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, 1987, 75-265. 
  27. [27] P. R. Heath, Product formulae for Nielsen numbers of fibre maps, Pacific J. Math. 117 (2) (1985), 267-289. Zbl0571.55002
  28. [28] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, AMS, 1983. 
  29. [29] B. Jiang, Estimation of the number of periodic orbits, Preprint of Universität Heidelberg, Mathematisches Institut, Heft 65, Mai 1993. 
  30. [30] B. Jiang and S. Wang, Lefschetz numbers and Nielsen numbers for homeomorphisms on aspherical manifolds, in: Topology - Hawaii, World Sci., Singapore, 1992, 119-136. Zbl1039.55502
  31. [31] S. Lang, Algebra, Addison-Wesley, 1993. 
  32. [32] A. A. Kirillov, Elements of the Theory of Representations, Springer Verlag, 1976. 
  33. [33] A. Mal'cev, On a class of homogeneous spaces, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 9-32 (in Russian). 
  34. [34] A. Manning, Axiom A diffeomorphisms have rational zeta function, Bull. London Math. Soc. 3 (1971), 215-220. Zbl0219.58007
  35. [35] J. Milnor, Infinite cyclic covers, in: Proc. Conf. 'Topology of Manifolds' in Michigan 1967, 115-133. 
  36. [36] J. Milnor, A duality theorem for the Reidemeister torsion, Ann. of Math. 76 (1962), 137-147. Zbl0108.36502
  37. [37] W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233-305. Zbl0395.57011
  38. [38] B. Norton-Odenthal, Ph. D. Thesis, University of Wisconsin, Madison, 1991. 
  39. [39] W. Parry and M. Pollicott, Zeta functions and the periodic structure of hyperbolic dynamics, Astérisque 187-188 (1990). Zbl0726.58003
  40. [40] D. Ray and I. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210. Zbl0239.58014
  41. [41] K. Reidemeister, Automorphism von Homotopiekettenringen, Math. Ann. 112 (1936), 586-593. 
  42. [42] G. de Rham, Complexes à automorphismes et homéomorphie différentiable, Ann. Inst. Fourier 2 (1950), 51-67. 
  43. [43] W. Rudin, Fourier Analysis on Groups, Interscience, 1962. 
  44. [44] D. Ruelle, Zeta function for expanding maps and Anosov flows, Invent. Math. 34 (1976), 231-242. Zbl0329.58014
  45. [45] M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175-179. 
  46. [46] H. Steinlein, Ein Satz über den Leray-Schauderschen Abbildungsgrad, Math. Z. 126 (1972), 176-208. Zbl0223.47023
  47. [47] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. AMS 19 (1988), 417-431. Zbl0674.57008
  48. [48] F. Wecken, Fixpunktklassen II, Math. Ann. 118 (1942), 216-234. 
  49. [49] A. Weil, Numbers of solutions of equations in finite fields, Bull. AMS 55 (1949), 497-508. Zbl0032.39402
  50. [50] P. P. Zabreĭko and M. A. Krasnosel'skiĭ, Iterations of operators and fixed points, Dokl. Akad. Nauk SSSR 196 (1971), 1006-1009 (in Russian). 

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