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2 Correct second paragraph

Poincaré Theorem on Kleinian groups (groups acting discontinously on Euclidean or hyperbolic spaces or on spheres) provides a method to obtain a presentation of a Kleinian group from a fundamental polyhedra.

I know the proof in Maskit book (Kleinian groups) but I would like to know other proofsand . I also know other proofs of for Fuchsian groups (dimension 2) which does not generalize to higher dimension (e.g. Beardon's book, The geometry of discrete groups).

I have two motivations: 1) Maskit proof also proves Poincaré Polyhedra Theorem, which states the necessary and sufficient conditions for a polyhedra to be fundamental polyhedra of some Kleinian group. I have the filling that a direct proof of the "presentation theorem" should be possible and simpler than proving the "Polyhedra Theorem".

2) Does Poincaré Theorem generalizes to direct product of hyperbolic spaces?

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# Poincaré Theorem on presentation from a fundamental polyhedra

Poincaré Theorem on Kleinian groups (groups acting discontinously on Euclidean or hyperbolic spaces or on spheres) provides a method to obtain a presentation of a Kleinian group from a fundamental polyhedra.

I know the proof in Maskit book (Kleinian groups) but I would like to know other proofs and other proofs of Fuchsian groups (dimension 2) which does not generalize to higher dimension.

I have two motivations: 1) Maskit proof also proves Poincaré Polyhedra Theorem, which states the necessary and sufficient conditions for a polyhedra to be fundamental polyhedra of some Kleinian group. I have the filling that a direct proof of the "presentation theorem" should be possible and simpler than proving the "Polyhedra Theorem".

2) Does Poincaré Theorem generalizes to direct product of hyperbolic spaces?