Dynamics of the Action of a CAT(0) Group on the Boundary
- Kim E. Ruane
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
The main theorem here gives a geometric condition on the fixed point sets of two hyperbolic isometries of a CAT(0) group which guarantees that the subgroup generated by the two elements contains a free subgroup. A result in section 3 shows that an element extends to the identity on the boundary if and only if the element is virtually central in the CAT(0) group. Finally, the two dimensional case is considered in the last section of the paper and there it is shown that powers of two hyperbolic isometries commute if and only if a geometric condition on the fixed point sets is satisfied.
- Alexandrov, A. D.: A theorem on triangles in a metric space and some applications, Trudy Mat. Inst. Steklov. 38 (1951), 5-23 (Russian).
- Ballman, W.: Axial isometries of manifolds of nonpositive curvature, Math. Ann. 259 (1982), 131-144.
- Ballman, W. and Brin, M.: Orbihedra of Nonpositive Curvature, Inst. Hautes Etudes Sci. Publ. Math. 82 (1995), 169-209.
- Ballman W., Gromov, M. and Schroeder, V.: Manifolds of Nonpositive Curvature, Birkhäuser, Basel, 1985.
- Bowditch, B.: Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), 145-186.
- Bowers, P. and Ruane K.: Boundaries of nonpositively curved groups of the form G × ℤ n, Glasgow Math. J. 38 (1996), 177-189.
- Bridson, M. R. and Haefliger, A.: Metric spaces of non-positive curvature, unpublished.
- Eberlein, P.: Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Math., University of Chicago Press, 1996.
- Gromov, M.: Hyperbolic groups, In: S. M. Gersten (ed.), Essays in Group Theory, Springer-Verlag, New York, 1987, 75-264.
- Kleiner, B.: Local structure of length spaces with curvature bounded above, to appear in Math. Zeit, 1997.
- Leeb, B.: A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Preprint, 1997.
- Lyndon, R. C. and Schupp, P. E.: Combinatorial Group Theory, Springer-Verlag, New York, 1977.
- Mihalik, M. and Ruane, K.: CAT(0) HNN-extensions with non-locally connected boundary, Topology Appl. 110(1) (2000), 83-98.
- Ontaneda, P.: Local extendability of CAT(0) spaces, Preprint (1996).
- Paulin, F.: Constructions of hyperbolic groups via hyperbolizations of polyhedra, In: E. Ghys, A. Haefliger, A. Verjovsky (eds.), Group Theory from a Geometrical Viewpoint, World Scientific, Singapore, 1991, pp. 313-328.
- Swarup, G.: On the cut point conjecture, Elec. Res. Announ. Amer. Math. Soc. 2 (1996), 98-100.
- Swenson, E.: Subgroups of negatively curved groups, Preprint (1993).
- Dynamics of the Action of a CAT(0) Group on the Boundary
Volume 84, Issue 1-3 , pp 81-99
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Kim E. Ruane (1)
- Author Affiliations
- 1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, U.S.A.