# 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. 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?

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.

• The last section in Ratcliffe's book is all about this. – Steve D Dec 22 '11 at 0:12

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.