Bounded cohomology and the Cheeger isoperimetric constant. (English) Zbl 1329.53061
Summary: We study equivalent conditions for the Cheeger isoperimetric constant of Riemannian manifolds to be positive. We first give a proof of Gromov’s assertion for locally symmetric spaces with infinite volume, which states that the existence of a bounded primitive of the Riemannian volume form is equivalent to the positivity of the Cheeger isoperimetric constant. Furthermore, under the assumption of pinched negative sectional curvature, we obtain another equivalent condition in terms of bounded cohomology classes. This generalizes T. Soma’s result [Duke Math. J. 88, No. 2, 357–370 (1997; Zbl 0880.57009)] for hyperbolic 3-manifolds to \(\mathbb R\)-rank one locally symmetric spaces.
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 53C35 | Differential geometry of symmetric spaces |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
Keywords:
bounded cohomology; Cheeger isoperimetric constant; geometrically finite manifold; bounded primitive; symmetric spaceCitations:
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