Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Some two-step and three-step nilpotent Lie groups with small automorphism groups


Author: S. G. Dani
Journal: Trans. Amer. Math. Soc. 355 (2003), 1491-1503
MSC (2000): Primary 22D45, 22E25; Secondary 22D40, 37D20
DOI: https://doi.org/10.1090/S0002-9947-02-03178-1
Published electronically: December 4, 2002
MathSciNet review: 1946401
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are ``small'' in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms.


References [Enhancements On Off] (What's this?)

  • 1. L. Auslander and J. Scheuneman, On certain automorphisms of nilpotent Lie groups, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif. 1968) pp. 9-15, Amer. Math. Soc., Providence, RI, 1970. MR 42:5284
  • 2. A. Bialynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203. MR 25:3936
  • 3. S.G. Dani, Nilmanifolds with Anosov automorphisms, J. London Math. Soc. (2) 18 (1978), 553-559. MR 80k:58082
  • 4. S.G. Dani, On automorphism groups acting ergodically on connected locally compact groups, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhya, Ser. A, 62 (2000), 360-366. MR 2001m:22013
  • 5. S.G. Dani, On ergodic ${\mathbb Z}^d$-actions on Lie groups by automorphisms, Israel J. Math. 126 (2001), 327-344. MR 2002j:37006
  • 6. S.G. Dani and M. McCrudden, A criterion for exponentiality in certain Lie groups, J. Algebra 238 (2001), 82-98. MR 2002b:22010
  • 7. K. Dekimpe, Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds, Trans. Amer. Math. Soc. 353 (2001), 2859-2877. MR 2002c:37043
  • 8. K. Dekimpe and W. Malfait, A special class of nilmanifolds admitting an Anosov diffeomorphism, Proc. Amer. Math. Soc. 128 (2000), 2171-2179. MR 2000m:37029
  • 9. J. Dixmier and W.G. Lister, Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc. 8 (1957), 155-158. MR 18:659a
  • 10. J. L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76 (1970), 52-56. MR 40:2789
  • 11. G. P. Hochschild, The Basic Theory of Algebraic Groups and Lie Algebras, Graduate Texts in Mathematics 75, Springer-Verlag, 1981. MR 82i:20002
  • 12. W. Malfait, Anosov diffeomorphisms on nilmanifolds of dimension at most six, Geometriae Dedicata (3) 79 (2000), 291-298. MR 2001h:20041
  • 13. M.S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, 1972. MR 58:22394a
  • 14. J.-P. Serre, Complex Semisimple Lie Algebras, Springer Monographs in Mathematics, Springer-Verlag, 2001 (reprinted edition). MR 2001h:17001
  • 15. V.S. Varadarajan, Lie Groups, Lie Algebras and their Representations, Graduate Texts in Mathematics 102, Springer-Verlag, 1984 (reprinted edition). MR 85e:22001
  • 16. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer-Verlag, 1982. MR 84e:28017

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22D45, 22E25, 22D40, 37D20

Retrieve articles in all journals with MSC (2000): 22D45, 22E25, 22D40, 37D20


Additional Information

S. G. Dani
Affiliation: Erwin Schrödinger Institute, Boltzmanngasse 9, A-1090 Vienna, Austria
Address at time of publication: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India
Email: dani@math.tifr.res.in

DOI: https://doi.org/10.1090/S0002-9947-02-03178-1
Received by editor(s): April 29, 2002
Received by editor(s) in revised form: July 12, 2002
Published electronically: December 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society