On discontinuous subgroups acting on solvable homogeneous spaces. (English) Zbl 1241.22009

This note deals with some features of the deformation space of properly discontinuous actions of a discrete subgroup on a homogeneous space, namely stability and rigidity. A classical local rigidity Theorem (Selberg and Weil, Kobayashi) asserts that there are no continuous deformations of cocompact discontinuous groups for \(G/H\) for a linear noncompact semisimple Lie group \(G\) except for few cases. When \(G\) is nilpotent it is conjectured here that the local rigidity property does not hold. When \(G\) is assumed to be exponential solvable and \(H\subset G\) is a maximal subgroup the author shows that the local rigidity property holds if and only if the group is isomorphic to the group of affine transformations of the line. The author also studied the notion of stable discontinuous subgroups following Kobayashi-Nasrin and he characterized stable discrete subgroups for Heisenberg groups. Unfortunately the detailed proofs cannot be found in this note.


22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
32G05 Deformations of complex structures
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[1] 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. · Zbl 1197.22003 · doi:10.1093/imrn/rnp193
[2] A. Baklouti and F. Khlif, Deforming discontinuous subgroups for threadlike homogeneous spaces, Geom. Dedicata 146 (2010), 117-140. · Zbl 1190.22009 · doi:10.1007/s10711-009-9429-3
[3] 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. · Zbl 0627.22012 · doi:10.1007/BF01391829
[4] T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), no. 2, 249-263. · Zbl 0662.22008 · doi:10.1007/BF01443517
[5] 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. · Zbl 1193.22010
[6] T. Kobayashi, On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geom. Phys. 12 (1993), no. 2, 133-144. · Zbl 0815.57029 · doi:10.1016/0393-0440(93)90011-3
[7] 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. · Zbl 0899.43005 · doi:10.1016/B978-012625440-2/50004-5
[8] T. Kobayashi, Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie Theory 6 (1996), no. 2, 147-163. · Zbl 0863.22010
[9] T. Kobayashi, Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds, Math. Ann. 310 (1998), no. 3, 395-409. · Zbl 0891.22014 · doi:10.1007/s002080050153
[10] T. Kobayashi, Discontinuous groups for non-Riemannian homogeneous spaces, in Mathematics unlimited–2001 and beyond , 723-747, Springer, Berlin. · Zbl 1023.53031 · doi:10.1007/978-3-642-56478-9_8
[11] 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. · Zbl 1124.57015 · doi:10.1142/S0129167X06003862
[12] 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. · Zbl 0201.36603
[13] A. Weil, On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578-602. · doi:10.2307/1970212
[14] A. Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149-157. · Zbl 0192.12802 · doi:10.2307/1970495
[15] 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. · Zbl 1151.22014
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