Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Expanding maps on infra-nilmanifolds of homogeneous type

Authors: Karel Dekimpe and Kyung Bai Lee
Journal: Trans. Amer. Math. Soc. 355 (2003), 1067-1077
MSC (2000): Primary 37D20; Secondary 17B30, 17B70
Published electronically: October 24, 2002
MathSciNet review: 1938746
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient $E\backslash L$, where $L$ is a connected and simply connected nilpotent Lie group and $E$is a torsion-free uniform discrete subgroup of $L {\mathbb o} C$, with $C$ a compact subgroup of $\operatorname{Aut}(L)$. We show that if the Lie algebra of $L$ is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.

References [Enhancements On Off] (What's this?)

  • 1. Dekimpe, K.
    Almost-Bieberbach Groups: Affine and Polynomial Structures, volume 1639 of Lecture Notes in Math.,
    Springer-Verlag, 1996. MR 2000b:20066
  • 2. Dekimpe, K. and Malfait, W.
    Affine structures on a class of virtually nilpotent groups.
    Topol. and its Applications, 1996, 73 (2), pp. 97-119. MR 97j:57060
  • 3. Gromov, M.
    Groups of polynomial growth and expanding maps.
    Inst. Hautes Études Sci. Publ. Math., 1981, 53, pp. 53-73. MR 83b:53041
  • 4. Hall, P.
    The Edmonton Notes on Nilpotent Groups.
    Queen Mary College Math. Notes, London, 1969. MR 44:316
  • 5. Johnson, R. W.
    Homogeneous Lie Algebras and Expanding Automorphisms.
    Proc. Amer. Math. Soc., 1975, 48 (2), pp. 292-296. MR 51:10417
  • 6. Kargapolov, M. and Merzljakov, J.
    Fundamentals of the Theory of Groups, volume 62 of Grad. Texts in Math.
    Springer-Verlag, 1979. MR 80k:20002
  • 7. Lee, H. and Lee, K. B.
    Expanding maps on 2-step infra-nilmanifolds.
    Topology Appl., 2002, 117 (1), pp. 45-58. MR 2002i:57038
  • 8. Lee, K. B. and Raymond, F.
    Rigidity of almost crystallographic groups.
    Contemporary Math. Amer. Math. Soc., 1985, 44, pp. 73-78. MR 87d:57026
  • 9. Leger, G.
    Derivations of Lie Algebras III.
    Duke Math. J., 1963, 30 (4), pp. 637-645. MR 28:3064
  • 10. Mal'cev, A. I.
    On a class of homogeneous spaces.
    Translations Amer. Math. Soc., 1951, 39, pp. 1-33. MR 12:589e
  • 11. Segal, D.
    Polycyclic Groups.
    Cambridge Tracts in Mathematics, No. 82, Cambridge University Press, 1983. MR 85h:20003

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37D20, 17B30, 17B70

Retrieve articles in all journals with MSC (2000): 37D20, 17B30, 17B70

Additional Information

Karel Dekimpe
Affiliation: Katholieke Universiteit Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium

Kyung Bai Lee
Affiliation: University of Oklahoma, Norman, Oklahoma 73019

Keywords: Infra-nilmanifold, expanding map, homogeneous Lie group
Received by editor(s): December 11, 2000
Received by editor(s) in revised form: March 15, 2002
Published electronically: October 24, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society