Ends of covering 3-manifolds

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Through the studies of ends of hyperbolic 3-manifolds, we obtained some results concering bounded cohomology.First, we showed, by using a certain hyperbolic 3-manifold, that the naturally defined pseudonorm on the third bounded cohomology H^3_(Z*Z ; R) is not a norm. As a corollary to this result, it is shown that, for any group G admitting a surjective homomorphism f : G*Z*Z,the pseudonorm on H^3_(G ; R) is not a norm.Next, we presented a rigidity theorem of certain hyperbolic 3-manifolds of infinite volume. Let SIGMA_g be a closed, connected, orientable surface of genus g>1. For any hyperbolic 3-manifold M homotopy-equivalent to SIGMA_g, the volume of M is infinite. Here, we consider the case where M has no geometrically finite ends, that is, M is doubly-degenerated. If the infimum inj(M) of injectivity radii at all points in M is positive, then by Minsky's Ending Lamination Theorem, the hyperbolic structure on M is determined only by its ending laminations. For any such M,M' with inj(M) >0, inj(M') > 0, we presented a condition equivalent to that M and M' have the same ending laminations in terms of the fundamental classes [omega_M], [omega_<M'>] defined as elements of H^3_(SIGMA_g ; R). Though [omega_M] = [omega_<M'>] is a sufficient condition for M isometric to M', we proved that a (formaly) weaker condition can be a necessary and sufficient condition for that.Furthermore, by using R.Canary's Covering Theorem, we showed that a topologically tame Kleinian group G is geometrically finite if and only if the funtametal class of G in H^3_(G ; R) is zero. As an application, we proved that, for any group G with a surjevtive homomorphism f : G*Z*Z,the dimension of H^3_(G ; R) is the cardinarity of continuum.