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We rigorously define the Liouville action functional for finitely generated,purely loxodromic quasi-Fuchsian group using homology and cohomology doublecomplexes naturally associated with the group action. We prove that theclassical action - the critical point of the Liouville action functional,considered as a function on the quasi-Fuchsian deformation space, is anantiderivative of a 1-form given by the difference of Fuchsian andquasi-Fuchsian projective connections. This result can be considered as globalquasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity.We prove that the classical action is a Kahler potential of the Weil-Peterssonmetric. We also prove that Liouville action functional satisfies holographyprinciple, i.e., it is a regularized limit of the hyperbolic volume of a3-manifold associated with a quasi-Fuchsian group. We generalize these resultsto a large class of Kleinian groups including finitely generated, purelyloxodromic Schottky and quasi-Fuchsian groups and their free combinations.Comment: 60 pages, proof of the Lemma 5.1 corrected, references and section  5.3 adde

Liouville Action and Weil-Petersson Metric on Deformation Spaces, Global Kleinian Reciprocity and Holography