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Bounded cohomology of closed surfaces. (English) Zbl 0881.57019
Bounded cohomology was introduced by M. Gromov in [Publ. Math., Inst. Hautes Étud. Sci. 56, 5-99 (1982; Zbl 0516.53046)]. The author investigates the third bounded cohomology group of a closed, connected surface of genus greater than one and shows how it relates to the geometry of hyperbolic 3-manifolds. One of the results is a version of the Gromov-Thurston-Mostow Rigidity Theorem which applies to a certain class of hyperbolic 3-manifolds with infinite volume. Previous studies of the bounded cohomology of surfaces have concentrated on the second bounded cohomology group, for example, R. Brooks and C. Series [Topology 23, 29-36 (1984; Zbl 0523.55011)], J. Barge and E. Ghys [Invent. Math. 92, No. 3, 509-526 (1988; Zbl 0641.55015)], and Y. Mitsumatsu [Topology 23, 465-471 (1984; Zbl 0568.55002)].
Reviewer: J.Hebda (St.Louis)

MSC:
57N65 Algebraic topology of manifolds
57M50 General geometric structures on low-dimensional manifolds
55N35 Other homology theories in algebraic topology
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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