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The anosov theorem for infranilmanifolds with an odd-order abelian holonomy group


We prove that for any continuous map of a given infranilmanifold with Abelian holonomy group of odd order. This theorem is the analogue of a theorem of Anosov for continuous maps on nilmanifolds. We will also show that although their fundamental groups are solvable, the infranilmanifolds we consider are in general not solvmanifolds, and hence they cannot be treated using the techniques developed for solvmanifolds.



  1. 1.

    Anosov DV: Nielsen numbers of mappings of nil-manifolds. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 1985,40(4(244)):133–134. English translation: Russian Math. Surveys, 40 (1985), pp. 149–150

    MathSciNet  Google Scholar 

  2. 2.

    Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Illinois; 1971:vi+186.

    Google Scholar 

  3. 3.

    Dekimpe K: The construction of affine structures on virtually nilpotent groups. Manuscripta Mathematica 1995,87(1):71–88. 10.1007/BF02570462

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Dekimpe K: Almost-Bieberbach Groups: Affine and Polynomial Structures, Lecture Notes in Mathematics. Volume 1639. Springer, Berlin; 1996:x+259.

    Google Scholar 

  5. 5.

    Heath PR, Keppelmann EC: Model solvmanifolds for Lefschetz and Nielsen theories. Quaestiones Mathematicae. Journal of the South African Mathematical Society 2002,25(4):483–501.

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Jezierski J, Kędra J, Marzantowicz W: Homotopy minimal periods for NR-solvmanifolds maps. Topology and its Applications 2004,144(1–3):29–49. 10.1016/j.topol.2004.02.018

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Jiang BJ: Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics. Volume 14. American Mathematical Society, Rhode Island; 1983:vii+110.

    Google Scholar 

  8. 8.

    Keppelmann EC, McCord CK: The Anosov theorem for exponential solvmanifolds. Pacific Journal of Mathematics 1995,170(1):143–159.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Kiang T-H: The Theory of Fixed Point Classes. Springer, Berlin; Science Press, Beijing; 1989:xii+174.

    Google Scholar 

  10. 10.

    Kwasik S, Lee KB: The Nielsen numbers of homotopically periodic maps of infranilmanifolds. Journal of the London Mathematical Society. Second Series 1988,38(3):544–554.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Lee KB: Maps on infra-nilmanifolds. Pacific Journal of Mathematics 1995,168(1):157–166.

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Lee JB, Lee KB: Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds. preprint, 2003

    Google Scholar 

  13. 13.

    Lee KB, Raymond F: Rigidity of almost crystallographic groups. In Combinatorial Methods in Topology and Algebraic Geometry (Rochester, NY, 1982), Contemp. Math.. Volume 44. American Mathematical Society , Rhode Island; 1985:73–78.

    Google Scholar 

  14. 14.

    Malfait W: The Nielsen numbers of virtually unipotent maps on infra-nilmanifolds. Forum Mathematicum 2001,13(2):227–237. 10.1515/form.2001.005

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to K Dekimpe.

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Dekimpe, K., De Rock, B. & Pouseele, H. The anosov theorem for infranilmanifolds with an odd-order abelian holonomy group. Fixed Point Theory Appl 2006, 63939 (2006).

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  • Differential Geometry
  • Computational Biology
  • Holonomy Group
  • Abelian Holonomy