Proceedings of the Japan Academy, Series A, Mathematical Sciences

On discontinuous subgroups acting on solvable homogeneous spaces

Ali Baklouti

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Abstract

We present in this note an analogue of the Selberg-Weil-Kobayashi local rigidity Theorem in the setting of exponential Lie groups and substantiate two related conjectures. We also introduce the notion of stable discrete subgroups of a Lie group $G$ following the stability of an infinitesimal deformation introduced by T. Kobayashi and S. Nasrin (cf. [11]). For Heisenberg groups, stable discrete subgroups are either non-abelian or abelian and maximal. When $G$ is threadlike nilpotent, non-abelian discrete subgroups are stable. One major aftermath of the notion of stability as reveal some studied cases, is that the related deformation spaces are Hausdorff spaces and in most of the cases endowed with smooth manifold structures.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 9 (2011), 173-177.

Dates
First available in Project Euclid: 4 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.pja/1320417396

Digital Object Identifier
doi:10.3792/pjaa.87.173

Mathematical Reviews number (MathSciNet)
MR2863361

Zentralblatt MATH identifier
1241.22009

Subjects
Primary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Secondary: 32G05: Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15]

Keywords
Solvable Lie subgroup proper action deformation space discontinuous subgroup rigidity stability

Citation

Baklouti, Ali. On discontinuous subgroups acting on solvable homogeneous spaces. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 9, 173--177. doi:10.3792/pjaa.87.173. https://projecteuclid.org/euclid.pja/1320417396


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References

  • A. Baklouti and I. Kédim, On non-abelian discontinuous subgroups acting on exponential solvable homogeneous spaces, Int. Math. Res. Not. IMRN 2010, no. 7, 1315–1345.
  • A. Baklouti and F. Khlif, Deforming discontinuous subgroups for threadlike homogeneous spaces, Geom. Dedicata 146 (2010), 117–140.
  • W. M. Goldman and J. J. Millson, Local rigidity of discrete groups acting on complex hyperbolic space, Invent. Math. 88 (1987), no. 3, 495–520.
  • T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), no. 2, 249–263.
  • T. Kobayashi, Discontinuous groups acting on homogeneous spaces of reductive type, in Representation theory of Lie groups and Lie algebras (Fuji-Kawaguchiko, 1990), 59–75, World Sci. Publ., River Edge, NJ, 1992.
  • T. Kobayashi, On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geom. Phys. 12 (1993), no. 2, 133–144.
  • T. Kobayashi, Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds, in Algebraic and analytic methods in representation theory (Sønderborg, 1994), 99–165, Perspect. Math., 17 Academic Press, San Diego, CA, 1997.
  • T. Kobayashi, Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie Theory 6 (1996), no. 2, 147–163.
  • T. Kobayashi, Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds, Math. Ann. 310 (1998), no. 3, 395–409.
  • T. Kobayashi, Discontinuous groups for non-Riemannian homogeneous spaces, in Mathematics unlimited–-2001 and beyond, 723–747, Springer, Berlin.
  • T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of $\mathbf{Z}^{k}$ on $\mathbf{R}^{k+1}$, Int. J. Math. 17 (2006), 1175-1190.
  • A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, in Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960), 147–164, Tata Inst. Fund. Res., Bombay, 1960.
  • A. Weil, On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578–602.
  • A. Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157.
  • T. Yoshino, Deformation spaces of compact Clifford-Klein forms of homogeneous spaces of Heisenberg groups, in Representation theory and analysis on homogeneous spaces, 45–55, RIMS Kokyuroku Bessatsu, B7 Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.