Bounded cohomology and 3-dimensional hyperbolic geometry

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Let H^3_ (SIGMA ; R) be the third bounded cohomology of a closed, orientable surface SIGMA of genus g>1. The head investigator proved that the pseudonorm ||・|| on H^3_ (SIGMA ; R) is not a norm by relying on the results in S.Matsumoto-S.Morita (1985). Moreover, by using a similar argument, we construct examples of the n-th bounded cohomology whose pseudonorm is not a norm for any n <greater than or equal> 5. They are the first examples showing that there exist bounded cohomologies without norm.For a topological space X,the subspace consisting of elements alpha of the k-th bounded cohomology H^k_ (X ; R) with ||alpha||=0 is called the zero-norm subspace of H^k_ (X ; R) and denoted by N^k (X). In this research, we investigated the third zero-norm subspace N^3 (SIGMA). The head investigator constructed non-trivial elements of N^3 (SIGMA) practically by using both a hyperbolic metric and a singular euclidean metric on SIGMA*R,where the euclidean metric is defined by using a measured lamination associated to a pseudo-Anosov automorphism of SIGMA. As an application of this practical construction, it was shown that the dimension of R-vector space N^3 (SIGMA) is the cardinality of continuum.Throughout the research of bounded cohomology, the head investigator obtained the notion of microchip decompositions on complexes consisting of hyperbolic 3-simplices. Later, it was turned out that the notion is useful also in investigating non-zero degree maps between 3-manifolds. In particular, if a non-zero degree map f : M*N from a closed 3-manifold to a hyperbolic 3-manifolds is given, one can define the structurc of a complex on M consisting of hyperbolic 3-simplices by using the hyperbolic structure on N.By using microchip decompositions on such complexes, it was proved that the number ofhyperbolic 3-manifolds admitting non-zero degree maps from a fixed M is finite.