DEFORMING THE R-FUCHSIAN (4,4,4)-TRIANGLE GROUP INTO A LATTICE MARTIN DERAUX Abstract. We prove that the last discrete deformation of the R- Fuchsian (4,4,4)-triangle group in PU(2, 1) is a cocompact arith- metic lattice. We also describe an experimental method for finding the combinatorics of a Dirichlet fundamental domain, and apply it to the lattice in question. 1. Introduction A lot of interest for complex hyperbolic geometry has been generated by Mostow's work in the late 1970's, exhibiting the first examples of nonarithmetic lattices in PU(n, 1) (in fact the current list of examples is only slightly larger, and all known examples in dimension four or higher are arithmetic). The major difficulty to construct such groups is to find efficient meth- ods for proving directly that a given group, for instance given by a number of generators, acts discretely on complex hyperbolic space. The construction of fundamental domains is much more complicated than in spaces with constant sectional curvature, since there are no totally geodesic real hypersurfaces. In particular there is no canonical choice for faces of a polyhedron, and the bare hands proofs of discreteness which have appeared to this day rely on using various kinds of hyper- surfaces, adapted to the situation at hand (bisectors in [11], C-spheres in [6], hybrid cones in [19], cones over totally geodesic subspaces in [5]).
- partial dirichlet domain
- ?2r? ?2r?
- tri- angle group
- fw has
- triangle group
- dirichlet domains
- angle condition
- ?r?
- hold only