Question

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. I also know other proofs 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?

Answer 1

Usually by Poincare Fundamental Polyhedron Theorem one means a collection of (preferably combinatorial and verifiable) condition ensuring that a polyderon in a hyperbolic space is the fundamental domain for a discrete group. Here are the sources:

1. In the real hyperbolic case in addition to the proof in Maskit's book there is the excellent paper [Epstein, David B. A.; Petronio, Carlo An exposition of Poincaré's polyhedron theorem. Enseign. Math. (2) 40 (1994), no. 1-2, 113--170].

2. There are also complex hyperbolic versions e.g. in [Falbel, Elisha; Zocca, Valentino A Poincaré's polyhedron theorem for complex hyperbolic geometry. J. Reine Angew. Math. 516 (1999), 133--158]

3. A very general version can be found in a recent preprint by Sasha Anan'in and Carlos H. Grossi here.

4. If memory serves, me this topic was also discussed extensively in [Ratcliffe, John G. Foundations of hyperbolic manifolds. Second edition. Graduate Texts in Mathematics, 149. Springer, New York, 2006].

5. I am not aware of any version specific for the product of hyperbolic spaces, but check 3 and 4. I have not been thinking of these matters since mid 90s, so I might have missed something.

Answer 2

If you only want a presentation theorem, I think this general result may be suitable:

A. M. Macbeath, Groups of homeomorphisms of a simply connected space, Ann. of Math. (2) 79 (1964), 473–488.