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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
##### Keywords:
boundary dimension; Gromov hyperbolic spaces
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##### References:
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