Abstract
A negatively curved or hyperbolic group, as introduced by M. Gromov, is a finitely generated group whose Cayley graphs asymptotically behave at infinity like a tree. Considering the action of a negatively curved group on one of its Cayley graphs, we study asymptotic directions in the set of conjugacy classes of this group. We then discuss some applications to group actions on ℝ — trees and manifolds of negative curvature.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. 124 (1986), 71–158.
F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), 139–162.
F. Bonahon, “Structures géométriques sur les variétés de dimension 3 et applications,” Thèse d’État, Université d’Orsay, 1985.
N. Bourbaki, “Éléments de mathématiques,” livre VI (Intégration), Hermann, Paris, 1965.
B. Bowditch, “Notes on Gromov’s hyperbolicity criterion for path-metric spaces,” manuscript, University of Warwick, 1989.
J. Cannon, W.P. Thurston, Group invariant Peano Curves, unpublished article (1982).
M. Coornaert, T. Delzant, A. Papadopoulos, “Notes sur les groupes hyperboliques de Gromov,” manuscript, Institut de Recherches Mathématiques Avancées, Strasbourg, 1989.
M. Culler, J.W. Morgan, Group actions on ℝ- trees, Proc. Lond. Math. Soc. 55 (1987), 571–604.
P. Eberlein, B. O’Neill, Visibility manifolds, Pac. J. Math. 46 (1973), 45–109.
W.J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), 205–218.
E. Ghys, P. de la Harpe et al., “Sur les groupes hyperboliques d’après Gromov,” (with contributions of W. Ballman, A. Haefliger, E. Salem, R. Strebel, M. Troyanov), manuscript, Ecole Normale Supérieure de Lyon and Université de Genève, 1989.
M. Gromov, Hyperbolic groups, in “Essays in group theory,” (S.M. Gersten ed.), Springer-Verlag, Berlin Heidelberg New York, 1987, pp. 75–263.
R.C. Lyndon, P.E. Schupp, “Combinatorial group theory,” Springer-Verlag, Berlin Heidelberg New York, 1977.
H. Masur, Measured foliations and handlebodies, Erg. Theory and Dyn. Syst. 6 (1986), 99–116.
J.W. Morgan, P.B. Shalen, Valuations,trees, and degenerations of hyperbolic structures, I, Ann. of Math. 120 (1984), 401–476.
J.W. Morgan, On Thurston’s uniformization theorem for the three-dimensional manifolds, in “The Smith Conjecture,” (H. Bass, J.W. Morgan ed.), Academic Press, 1984.
J. Alonso, T. Brady, D. Cooper, T. Delzant, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, H. Short, “Notes on negatively curved groups,” manuscript, Mathematical Sciences Research Institute, Berkeley, 1989.
J.-P. Otal, “Courants géodésiques et surfaces,” Thèse d’État, Université de Paris-Sud, Orsay, 1989.
F. Paulin, The Gromov topology on R- trees,Topol. Appl. 32 (1989), 197–221. [Ser] J.-P. Serre, “Arbres et amalgames,” Astérisque, Société Mathématique de France, 1977.
K. Sigmund, Generic properties of invariant measures for Axiom A- diffeomorphisms, Invent. Math. 11 (1970), 99–109.
K. Sigmund, On dynamical systems with the Specification Property, Trans. Amer. Math. Soc. 190 (1974), 285–299.
R. Skora, Geometric actions of surface groups on A -trees, preprint. [Sko2] R. Skora, Splittings of surfaces, preprint.
D. Sullivan, Cycles for the dynamical study of foliated spaces and complex manifolds, Invent. Math. 36 (1976), 225–255.
W.P. Thurston, “The topology and geometry of 3-manifolds,” Lecture notes, Princeton University, 1976–79.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Bonahon, F. (1991). Geodesic currents on negatively curved groups. In: Alperin, R.C. (eds) Arboreal Group Theory. Mathematical Sciences Research Institute Publications, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3142-4_5
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3142-4_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7811-5
Online ISBN: 978-1-4612-3142-4
eBook Packages: Springer Book Archive