Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits
Keywords:Affine nilpotent groups, Homogeneous manifolds, Proper actions, Properly discontinuous actions, Simply connected nilpotent Lie groups, Compact isotropy property (CI), Eigenvalues.
Criterion for proper actions has been established for a homogeneous space of reductive type by Kobayashi (Math. Ann. 1989, 1996). On the other hand, an analogous criterion to Kobayashis equivalent conditions was proposed by Lipsman (1995) for a nilpotent Lie group . Lipsman's Conjecture: Let be a simply connected nilpotent Lie group. Then the following two conditions on connected subgroups and are equivalent: (i) the action of on is proper; (ii) is compact for any The condition (i) is important in the study of discontinuous groups for the homogeneous space , while the second condition (ii) can easily be checked. The implication (i) (ii) is obvious, and the opposite implication (ii) (i) was known only in some lower dimensional cases. In this paper we prove the equivalence (i) ? (ii) for certain affine nilpotent Lie groups .
Keywords: Affine nilpotent groups; Homogeneous manifolds; Proper actions; Properly discontinuous actions; Simply connected nilpotent Lie groups; Compact isotropy property (CI); Eigenvalues.
© 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.
doi: http://dx.doi.org/10.3329/jsr.v4i2.7889 J. Sci. Res. 4 (2), 315-326 (2012)
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