【摘要】 这组文章,发展了拟有限生成的Klein群的解析理论,这种Klein群通常可能是无限生成的.若一个Klein群是拟有限生成的,它可表示为Γ=（γ₁,…,γ_n,Γ（B））,这里Γ（B）是Γ的极大的零化子群,本文研究了拟有限生成的Klein群的许多问题,如:有限性定理,面积定理,上同调,Poincare级数,及尖点估计等。在Ⅰ中,简单地回顾了有限生成的Klein群的若干结果,特别是Ahlfors有限性定理,这一定理是Klein群的解析理论的基石.其思想来源于Ahlfors文的证明之中. 在Ⅱ中,研究了Klein群的Ⅱ_2q-2-上同调的结构,引入了许多新的概念,如零化子空间,零化子群,Kra变换,Kra泛函,相对边缘子空间,q-代数扩张,代数扩张等.这一节的内容是研究拟有限生成的Klein群的基础。在Ⅲ中,引入了拟有限生成的Klein群的概念,并且得到拟有限生成的Klein群的有限性定理,面积定理及若干面积不等式。在Ⅳ中,引入了相对的Eichler积分空间,得到了拟有限生成的Klein群的一阶上同调的分解.并且研究了拟有限生成的Klein群的Poincare级数及尖点估计的理论.这一部分内容是Kra的推广。最后提出了一些理论中尚未解决的问题。更多还原

【Abstract】 In these series of papers, the authors developed the analytictheory of quasi-finitely generated Kleinian groups which may be usualinfinitely generated, and considered that a Kleinian group Γ is quasi-finitely generated if it can be represented by Γ=<γ₁, …, γ_n, Γ（B）>,Where Γ （B） is a maximal annihilated subgroup.They studied manyproblems of quasi-finitely generated Kleinian groups such as finitenesstheorem, area theorem, cohomology, Poincare series and estimations ofcusps etc.In I, they simplely recalled some results of finitely generated Kleiniangroups, particulurly Ahlfors’ finiteness theorem which is foundation ofthe analytic theory of Kleinian groups, and the authors ideas come fromthe proof of Ahlfors’ original paper.In Ⅱ, the authors studied structions of Π_2q-2 cohomology of Kleiniangroups, and intruduced some new notations such as annihilated subspaces,annihilated subgroups, Kra transformation, Kra functional, relativeboundary subspaces, q-algebric extension, algebric extension etc.In Ⅲ,they intruduced quasi-finitely generated Kleinian groups, andobtained finiteness theorem, area theorem and some area inequalities.In Ⅳ, they intruduced relative Eichler integral spaces, obtained decom-position of H¹, and also investigated the Poincare series of quasi-finitelygenerated Kleinian groups and the theory of estimations of cusps. Theresults of this part are extensions of Kra.Finally, the authors give some open problems on this theory.更多还原