Ends of covering 3-manifolds

通过对双曲型3流形端点的研究，得到了关于有界上同调的一些结果。首先，利用某双曲3流形证明了在第三有界上同调H^3_（Z*Z； R）上的自然定义伪范数不是范数。作为这个结果的一个推论，证明了对于任何群G承认一个满同态f: G*Z*Z，在H^3_（G； R）上的伪范数不是范数。其次，给出了一类无限体积双曲型3流形的刚性定理。设SIGMA_g为g>1属的一个闭合、连通、可定向曲面。对于任何与SIGMA_g同伦等价的双曲3流形M， M的体积是无限大的。这里，我们考虑M没有几何有限端点的情况，即M是双简并的。如果M上所有点的注入半径的最小注入(M)为正，则根据Minsky的结束层积定理，M上的双曲结构仅由其结束层积决定。对于任意这样的M，M‘具有inj(M) >, inj（M’） >，我们给出了一个等价的条件，即M和M‘在定义为H^3_（SIGMA_g； R）元素的基本类[omega_M]， [omega_<M’>]方面具有相同的结束层。虽然[omega_M] = [omega__ <M‘>]是M与M’等距的充分条件，但我们证明了一个（形式上）较弱的条件可以是它的充分必要条件。进一步，利用r.c arary的覆盖定理，证明了拓扑单调的Kleinian群G是几何有限的当且仅当H^3_（G； R）中的G的基类为零。作为一个应用，我们证明了对于任何具有生存同态f: G*Z*Z的群G， H^3_（G； R）的维数是连续统的基数。

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.