The Discrete Isometric Subgroups Of Complex Hyperbolic Space

Complex hyperbolic geometry is an important branch of modern mathemat-ics,and it keeps close relationship with complex analysis,differentiable manifold,moduli space and discrete group.Complex hyperbolic manifolds have rich geom-etry structures,such as complex structure,symplectic structure,and sphere CR structure etc.It can help us understand the geometric structure of hyperbolic manifolds by studying the deformation of discrete groups,so the representation and deformation of complex hyperbolic discrete groups is an important issue.R.E.Schwartz considers the representation of triangle group onto PU(2,1)in a series of his important papers,and proves that many hyperbolic manifolds have complete sphere CR structure.Firstly this paper considers the deformation problem of a family of triangle groups on complex hyperbolic space,which can be parameterized by Cartan an gular invariant,by deforming the group which generated by three R-cicles only with rotational symmetry of order 3,and constructing fundamental domain by R-spheres,then get a interview on which the groups are discrete and faithful.In 2001 R.E.Schwartz gave the following conjecture:let I1,I2,I3 be re-flections,if T=<I1,I2,I3>is a complex hyperbolic(p,q,r)triangle group,then complex hyperbolic representation T of(p,q,r)is discrete and faithful if and only if WA=I1I3I2I3 and WB=I1I2I3 are not elliptic elements.This paper assumes that I1,I2,I3 are complex reflections with order 2,considers the representa-tion of(3,4,?)triangle group onto PU(2,1),and gets a sufficient and necessary condition that these complex hyperbolic triangle groups are discrete and faithful representation of(3,4,?),then proves that in this case the conjecture of R.E.Schwartz is correct.The last work of this paper is that we estimate the lower bound for the volume of quaternionic hyperbolic orbifolds,and get a dominant expression only depends on the space dimensionality and the order of elliptic element.