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Isometric Group Actions on Hilbert Spaces: Structure of Orbits

Published online by Cambridge University Press:  20 November 2018

Yves de Cornulier ,
Romain Tessera  and
Alain Valette
Yves de Cornulier
Affiliation:
IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France e-mail: yves.decornulier@univ-rennes1.fr
Romain Tessera
Affiliation:
Department of Mathematics, Stevenson Center, Vanderbilt University, Nashville, TN 37240 e-mail: romain.a.tessera@vanderbilt.edu
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Abstract

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Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

Keywords

22D1043A3520F69affine actionsHilbert spacesminimal actionsnilpotent groups

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 5 , October 2008 , pp. 1001 - 1009
Copyright
Copyright © Canadian Mathematical Society 2008

References

[CTV] Cornulier, Y. de, Tessera, R., and Valette, A., Isometric group actions on Hilbert spaces: growth of cocycles . Geom. funct. Anal. 17(2007), no. 3, 770–792.Google Scholar
[Fe] Fell, J. M. G., Weak containment and induced representations of groups . Canad. J. Math. 14(1962), 237–268.Google Scholar
[Gu1] Guichardet, A., Sur la cohomologie des groupes topologiques. II . Bull. Sci. Math. 96(1972), 305–332.Google Scholar
[Gu2] Guichardet, A., Cohomologie des groupes topologiques et des alg`ebres de Lie. Textes Mathématiques 2, CEDIC, Paris, 1980.Google Scholar
[HiKa] Higson, N., and Kasparov, G., Operator K-theory for groups which act properly and isometrically on Hilbert space . Electron. Res. Announc. Amer. Math. Soc. 3(1997), 131–142.Google Scholar
[Ma] Martin, F., Analyse harmonique et 1-cohomologie réduite des groupes localement compacts. PhD Thesis, Univ. Neuchˆatel, Switzerland, 2003.Google Scholar
[Sh1] Shalom, Y., Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group . Ann. of Math. 152(2000), no. 1, 113–182.Google Scholar
[Sh2] Shalom, Y., Rigidity of commensurators and irreducible lattices . Invent. Math. 141(2000), no. 1, 1–54.Google Scholar
[Sh3] Shalom, Y., Harmonic analysis, cohomology, and the large scale geometry of amenable groups . Acta Math. 192(2004), no. 2, 119–185.Google Scholar