Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-17T21:01:43.297Z Has data issue: false hasContentIssue false
  • English
  • Français

Averaging Formula for Nielsen Numbers of Maps on Infra-Solvmanifolds of Type (R)

Published online by Cambridge University Press:  11 January 2016

Jong Bum Lee  and
Kyung Bai Lee
Jong Bum Lee
Affiliation:
Department of Mathematics Sogang University, Seoul 121-742, Korea, jlee@sogang.ac.kr
Kyung Bai Lee
Affiliation:
Department of Mathematics University of Oklahoma Norman, OK 73019, U.S.A., kblee@math.ou.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-solvmanifolds of type (R): Let M be such a manifold with holonomy group Ψ and let f: M → M be a continuous map. The averaging formula for Nielsen numbers

is proved. This is a workable formula for the difficult number N(f).

Keywords

55M2057S30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 196 , 2009 , pp. 117 - 134
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[1] Anosov, D. V., The Nielsen numbers of maps of nil-manifolds, Uspehi Mat. Nauk, 40 (1985), 133–134; Russian Math. Survey, 40 (1985), 149–150.Google Scholar
[2] Baues, O., Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology, 43 (2004), 903–924.CrossRefGoogle Scholar
[3] Brooks, R. B. S. Brown, R. F., Pak, J. and Taylor, D. H., Nielsen numbers of maps of tori, Proc. Amer. Math. Soc., 52 (1975), 398–400.CrossRefGoogle Scholar
[4] Dekimpe, K., Rock, B. De and Malfait, W., The Nielsen numbers of Anosov diffeo-morphisms on flat Riemannian manifolds, Forum Math., 17 (2005), 325–341.CrossRefGoogle Scholar
[5] Dekimpe, K., Rock, B. De and Malfait, W., The Anosov relation for Nielsen numbers of maps of infra-nilmanifolds, Monatsh. Math., 150 (2007), 1–10.CrossRefGoogle Scholar
[6] Dekimpe, K., Lee, K. B. and Raymond, F., Bieberbach theorems for solvable Lie groups, Asian J. Math., 5 (2001), 499–508.CrossRefGoogle Scholar
[7] Ha, K. Y. and Lee, J. B., Left invariant metrics and curvatures on simply connected three dimensional Lie groups, Math. Nachr., 282 (2009), 868–898.CrossRefGoogle Scholar
[8] Jezierski, J., Kȩdra, J. and Marzantowicz, W., Homotopy minimal periods for NR-solvmanifolds maps, Topology Appl., 144 (2004), 29–49.CrossRefGoogle Scholar
[9] Jiang, B., Lectures on Nielsen fixed point theory, Contemporary Mathematics Amer. Math. Soc., 14, 1983.Google Scholar
[10] Kamishima, Y., Lee, K. B. and Raymond, F., The Seifert construction and its applications to infranilmanifolds, Quart. J. Math. Oxford Ser. (2), 34 (1983), 433–452.CrossRefGoogle Scholar
[11] Kim, S. W., Lee, J. B. and Lee, K. B., Averaging formula for Nielsen numbers, Nagoya J. Math., 178 (2005), 37–53.CrossRefGoogle Scholar
[12] Keppelmann, E. and McCord, C., The Anosov theorem for exponential solvmanifolds, Pacific J. Math., 170 (1995), 143–159.CrossRefGoogle Scholar
[13] Kwasik, S. and Lee, K. B., The Nielsen numbers of homotopically periodic maps of infranilmanifolds, J. London Math. Soc. (2), 38 (1988), 544–554.CrossRefGoogle Scholar
[14] Lee, J. B. and Lee, K. B., Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006), 2011–2023.CrossRefGoogle Scholar
[15] Lee, K. B., Nielsen numbers of periodic maps on solvmanifolds, Proc. Amer. Math. Soc., 116 (1991), 575–579.CrossRefGoogle Scholar
[16] Lee, K. B., Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157–166.CrossRefGoogle Scholar
[17] Lee, K. B., Infra-solvmanifolds of type (R), Quart. J. Math. Oxford Ser. (2), 46 (1995), 185–195.CrossRefGoogle Scholar
[18] McCord, C., Nielsen numbers and Lefschetz numbers on solvmanifolds, Pacific J. Math., 147 (1991), 153–164.CrossRefGoogle Scholar
[19] Raghunathan, M. S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer-Verlag, 1972.Google Scholar
[20] Saito, M., Sur certains groupes de Lie resolubles, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 7 (1957) 1–11.Google Scholar
[21] Saito, M., Sur certains groupes de Lie resolubles II, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 7 (1957) 157–168.Google Scholar
[22] Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401–487.CrossRefGoogle Scholar
[23] Vinberg, E. B., Gorbatsevich, V. V. and Shvartsman, O. V., Discrete subgroups of Lie groups, Current Problems in Mathematics. Fundamental Directions, vol. 21, Itogi Nauki i Tekhniki, vol. 215, Akad. Nauk SSSR, VINITI, Moscow, 1988, pp. 5–120 (in Russian).Google Scholar