Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 020, 17 pages      arXiv:0902.2977      https://doi.org/10.3842/SIGMA.2009.020

Discrete Cocompact Subgroups of the Five-Dimensional Connected and Simply Connected Nilpotent Lie Groups

Amira Ghorbel and Hatem Hamrouni
Department of Mathematics, Faculty of Sciences at Sfax, Route Soukra, B.P. 1171, 3000 Sfax, Tunisia

Received July 16, 2008, in final form February 09, 2009; Published online February 17, 2009

Abstract
The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if G = N × A is a connected, simply connected, nilpotent Lie group with an Abelian factor A, then every uniform subgroup of G is the direct product of a uniform subgroup of N and Zr where r = dim A.

Key words: nilpotent Lie group; discrete subgroup; nil-manifold; rational structures, Smith normal form; Hermite normal form.

pdf (294 kb)   ps (205 kb)   tex (19 kb)

References

  1. Auslander L., Green L., Hahn F., Flows on homogeneous spaces, Annals of Mathematics Studies, no. 53, Princeton University Press, Princeton, N.J., 1963.
  2. Corwin L., Greenleaf F.P., Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples, Cambridge Studies in Advanced Mathematics, Vol. 18, Cambridge University Press, Cambridge, 1990.
  3. Dixmier J., Sur les représentations unitaires des groupes de Lie nilpotents. III, Canad. J. Math. 10 (1958), 321-348.
  4. Gorbatsevich V., About existence and non-existence of lattices in some solvable Lie groups, Preprint ESI, no. 1253, 2002.
  5. Gordon C.S., Wilson E.N., The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), 253-271.
  6. Hamrouni H., Discrete cocompact subgroups of the generic filiform nilpotent Lie groups, J. Lie Theory 18 (2008), 1-16.
  7. Hungerford T.W., Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York - Berlin, 1974.
  8. Lauret J., Will C.E., On Anosov automorphisms of nilmanifolds, J. Pure Appl. Algebra 212 (2008), 1747-1755.
  9. Malcev A.I., On a class of homogeneous spaces, Amer. Math. Soc. Translation, no. 39, 1951.
  10. Matsushima Y., On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J. 2 (1951), 95-110.
  11. Milnes P., Walters S., Discrete cocompact subgroups of the four-dimensional nilpotent connected Lie group and their group C*-algebras, J. Math. Anal. Appl. 253 (2001), 224-242.
  12. Milnes P., Walters S., Discrete cocompact subgroups of G5,3 and related C*-algebras, Rocky Mountain J. Math. 35 (2005), 1765-1786, math.OA/0105104.
  13. Milnes P., Walters S., Simple infinite dimensional quotients of C*(G) for discrete 5-dimensional nilpotent groups G, Illinois J. Math. 41 (1997), 315-340.
  14. Onishchik A.L., Vinberg E.B., Lie groups and Lie algebra. II. Discrete subgroups of Lie groups and cohomologies of Lie groups and Lie algebras, Encyclopaedia of Mathematical Sciences, Vol. 21, Springer-Verlag, Berlin, 2000.
  15. Pesce H., Calcul du spectre d'une nilvariété de rang deux et applications, Trans. Amer. Math. Soc. 339 (1993), 433-461.
  16. Raghunathan M.S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York - Heidelberg, 1972.

Previous article   Next article   Contents of Volume 5 (2009)