Abstract
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of “nonelementary” discrete isometry groups of higher rank symmetric spaces. We obtain the natural generalisations of some well-known results from Kleinian group theory. Our main tool consists in a detailed description of the dynamics of individual isometries. As a by-product, we give a new geometric construction of free isometry groups with parabolic elements in higher rank symmetric spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ballmann W. (1995) Lectures on Spaces of Nonpositive Curvature, DMV Seminar, Band Vol. 25. Birkhäuser, Basel
Ballmann W., Gromov M., Schroeder V. (1985) Manifolds of Nonpositive Curvature, Progr. Math. vol. 61. Birkhäuser, Boston MA
Benoist Y. (1997) Propriétés asymptotiques des groupes linéaires I. Geom. Funct. Anal. 7, 1–47
Conze J. P., Guivarc’h Y. (2000) Limit Sets of Groups of Linear Transformations. Ergodic Theory Harmonic Anal. Sankhy a Ser. A 62(3): 367–385
Dal’bo, F., Kim, I.: Ergodic geometry on the product of Hadamard manifolds, Preprint (2000)
Eberlein, P.: Geometry of Non-Positively Curved Manifolds. Chicago Lectures in Mathematics, Chicago Univ. Press, Chicago (1996)
de la Harpe P. (1983) Free Groups in Linear Groups. Enseign. Math. 29, 129–144
Helgason S. (1978) Differential Geometry, Lie groups, and Symmetric Spaces. Academic Press, New York
Link, G.: Limit Sets of Discrete Groups acting on Symmetric Spaces, www.ubka.uni- karlsruhe.de/cgi-bin/psview? document=2002/mathematik/9, Dissertation, Karlsruhe (2002)
Mostow, G. D.: Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematics Studies, no. 78. Princeton University Press, Princeton N J (1973)
Parreau A. (2000) Dégénérescences de sous-groupes discrets de groupes de Lie semisimples et actions de groupes sur des immeubles affines. Thèse de doctorat, Orsay
Warner G. (1972) Harmonic Analysis on Semisimple Lie Groups I. Springer-Verlag, Berlin, Heidelberg, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Link, G. Geometry and dynamics of discrete isometry groups of higher rank symmetric spaces. Geom Dedicata 122, 51–75 (2006). https://doi.org/10.1007/s10711-006-9090-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-006-9090-z