Abstract
We prove that, in some situations, an induced action from a normal subgroup preserves a geometric structure. Combined with known geometric rigidity results, this result implies certain rigidity statements concerning the full diffeomorphism group of a manifold. It also provides many examples of actions on Lorentz manifolds. Combining these with a small number of well-known actions, we get the full list of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold. We give an example of a connected nilpotent Lie group with no complicated action on a Lorentz manifold. We show that, if a connected Lie group has a normal closed subgroup isomorphic to a (two-dimensional) cylinder, then it admits a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.
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References
Adams, S.:Another proof of Moore's ergodicity theorem for SL(2;R), Contemp.Math.215 (1998),183–187.
Adams, S.:Transitive actions on Lorentz manifolds with noncompact stabilizer.Preprint,1999.
Adams, S.:Locally free actions on Lorentz manifolds,Geom.Funct.Anal. 10 (2000),453–515.
Adams, S.:Isometric actions of SO(2) ⋉ ℝ2.to appear in Proc.Conf.of Besse: Round Table on Global Pseudo-Riemannian Geometry.
Adams, S.:Orbit nonproper actions on Lorentz manifolds.to appear,Geom. Funct.Anal.
Adams, S.:Orbit nonproper dynamics on Lorentz manifolds.to appear, Illinois J.Math.(2002).
Adams, S.and Stuck, G.:Conformal actions of SL n (ℝ) and SL (n(ℝ) ⋉ ℝnon Lorentz manifolds,Tran.Amer.Math.Soc. 352 (2000), 3913–3936.
Adams, S.and Stuck, G.:Isometric actions of SLn (ℝ) x ℝ n on Lorentz manifolds,Israel J.Math.121 (2001),93–111.
Arveson, W.:Invitation to C* –Algebras,Springer-Verlag, New York,1976.
Ellis, R.and Nerurkar, M.:Weakly almost periodic flows,Trans.Amer.Math. Soc. 313(1)(1989),103–119.
Helgason, S.:Differential Geometry,Lie Groups,and Symmetric Spaces. Academic Press, New York,1978.
Varadarajan, V.S.:Lie Groups,Lie Algebras,and Their Representations. Springer-Verlag, New York,1974.
Zimmer, R.J.:Ergodic Theory and Semisimple Groups,Birkhäuser, Boston, 1984.
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Adams, S. Induction of Geometric Actions. Geometriae Dedicata 88, 91–112 (2001). https://doi.org/10.1023/A:1013191613349
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DOI: https://doi.org/10.1023/A:1013191613349