Characterizations of higher rank hyperbolicity. (English) Zbl 1535.53043
Gromov hyperbolicity can be seen as a natural generalization of the concept of a simply connected manifold with negative curvature in the quasi-isometric setting. Analogous to that, this article provides a quasi-isometric generalization of the notion of simply connected manifolds with non-positive curvature. Just like the case of Gromov hyperbolicity, this notion can be characterized via isoperimetric inequalities, slim simplex properties, the Morse lemma and bounded filling radius. The article shows that these characterizations are all equivalent. One of the equivalences, namely the implication from bounded rank to a linear isoperimetric inequality, was a long standing open problem.
Reviewer: Chenxi Wu (Piscataway)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 51F30 | Lipschitz and coarse geometry of metric spaces |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
Keywords:
Gromov hyperbolicity; non-positive curvature; quasi-minimizer; Morse lemma; asymptotic rank; filling radius; linear isoperimetric inequalityReferences:
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