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Boundary dimension in negatively curved spaces. (English) Zbl 0837.53037
The paper deals with the so-called Gromov hyperbolic (negatively curved) spaces \(X\) which generalize the most important case of negatively curved groups, see M. Gromov [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. Gromov’s condition on \(X\) to have finite-dimensional boundary \(\partial X\) is that \(X\) be a simplicial graph with bounded valence and unit length edges (which cover the case of the Cayley graph of a hyperbolic group, see also Prop. 6.2 in E. Ghys and P. de la Harpe [Sur les groupes hyperboliques d’après Mikhael Gromov, Prog. Math. 83, Boston, MA: Birkhäuser (1990; Zbl 0731.20025)].
The author gives a more general result which implies that such \(X\) with a cocompact isometry group action always has finite-dimensional boundary (the author’s usage compact does not imply Hausdorff). As another result, the author gives an example of a negatively curved space \(X\) with infinite-dimensional boundary.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20F65 Geometric group theory
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[1] Alonso, J., Brady, T., Cooper, D., Delzant, T., Ferlini, T., Lustig, M., Mihalik, M., Shapiro, M., and Short, H.: Notes on word hyperbolic groups, in E. Ghys, A. Atteafliger and A. Verjovsky (eds),Group Theory from a Geometrical Viewpoint, World Scientific, Singapore, 1992. · Zbl 0849.20023
[2] Bestvina, M. and Mess, G.: The boundary of negatively curved groups, Preprint, 1990. · Zbl 0767.20014
[3] Cannon, J.: The theory of negatively curved spaces and groups, in Tim Bredford and Caroline Series (eds),Hyperbolic Geometry and Ergodic Theory, Oxford University Press, 1991, pp. 315-369. · Zbl 0764.57002
[4] Cannon, J. and Swenson, E.: Recognizing constant curvature discrete groups in dimension 3. Preprint, 1993. · Zbl 0910.20024
[5] Coornaert, M., Delzant, T. and Papadopoulos, A.T.:Gèometrie et thèorie des groupes, Lecture Notes in Math. 1441, Springer-Verlag, New York, 1991. · Zbl 0727.20018
[6] Epstein, D. with Cannon, J., Holt, D., Levy, S., Paterson, M. and Thurston, W.:Word Processing in Groups, Jones and Bartlett Publishers, Boston, 1992.
[7] Ghy, E. and De la Harpe, P.:Sur les groupes hyperboliques d’après Mikael Gromov, Progress Math. 83, Birkhaüser, Zürich, 1990.
[8] Gromov, M.: Hyperbolic groups, in S. Gersten (ed.):Essays in Group Theory MSRI Publication, Vol. 8, Springer-Verlag, New York, 1987. · Zbl 0634.20015
[9] Swenson, E.: Negatively curved groups and related topics, PhD Thesis, Brigham Young University, Provo, Utah, 1993.
[10] Swenson, Hyperbolic elements in negatively curved groups,Geom. Dedicata 55(2) (1995), 199-210. · Zbl 0834.20041 · doi:10.1007/BF01264930
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