形变空间上的Liouville作用量和Weil-Petersson度量，整体Kleinian互易性和全息
Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography
摘要：
利用与群作用相关的同调和上同调二重复形，严格定义了纯斜驶拟富克斯群的Liouville作用泛函.证明了Liouville作用泛函的经典作用量--作为拟Fuchsian变形空间上的函数的Liouville作用泛函的临界点是由Fuchsian和拟Fuchsian射影联络的差给出的1-形式的反导数.这一结果可以被认为是全局的拟Fuchsian互易性，它隐含了McMullen的拟Fuchsian互易性。我们证明了经典作用量是Weil-Petersson度量的Kahler势。我们还证明了Liouville作用泛函满足全息原理，即，它是与拟富克斯群相关的三维流形的双曲体积的正则化极限。我们将这些结果推广到一个大类的Kleinian群，包括生成的，纯斜驶Schottky和拟富克斯群和它们的自由组合。
英文摘要：
We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kahler potential of the Weil-Petersson metric. We also prove that Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups and their free combinations.
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