On Uniformizable Sphere CR Structures Of Hyperbolic3-manifolds

Let M be a complex hyperbolic 3-manifolds and we say M admits a uniformizable spherical CR structure if M is homeomorphic to ? \ ? where ? is a complex hyperbolic discrete subgroup and ? is the discontinuous domain of ?.It is a crucial problem in complex hyperbolic geometry whether a 3-manifolds admits a uniformizable spherical CR structure.This problem is very difficult for general 3-manifolds.So far,only a few hyperbolic 3-manifolds have been proved to admit a uniformizable spherical CR structure.This paper constructs Dirichlet fundamental domain of(3,3,5)-group.We find that combinatorics of this Dirichlet fundamental domain is similar to that construction proposed by M.Deraux and E.Falbel.Using their similar analytical methods,we analyze the combination of the fundamental domain on the boundary of complex hyperbolic space and prove m009 admitting a uniformizable spherical CR structure.Unlike the method used by M.Deraux and M.Acosta,our method gives the CR triangulation of m009 directly.