Deformation Of Complex Hyperbolic Triangle Groups And Conjugate Classes Of Torsion In Picard Modular Groups

Complex hyperbolic space is the unit ball in complex vector space with Bergman metric,and complex hyperbolic group is the holomorphic isometry group acting on the complex hyperbolic space.Complex hyperbolic groups have rich connections with complex geometry,low-dimensional topology,dynamical systems,etc.This thesis studies the deformation of complex hyperbolic triangle groups and conjugate classes of torsion in Picard modular groups.A very important kind of complex hyperbolic groups are complex hyperbolic triangle groups.Given three complex lines on the complex hyperbolic plane HC2,we assume the angles between each two complex lines are π/p,π/q and π/r respectively.Then the group(I1,I2,I3)generated by mirror reflections of these three complex lines is called a complex hyperbolic(p,q,r)triangle group.In this paper,we study the deformation problem of complex hyperbolic(3,n,∞)triangle groups,and obtain the discrete and faithful representation space.In precisely,we prove that when n>5,a complex hyperbolic(3,n,∞)triangle group is discrete and faithful if and only if/I1I3I2I3 is not elliptic.This result answers the conjecture introduced by Schwartz in 2002 about the discreteness and faithfulness of complex hyperbolic triangle groups in the case(3,n,∞).Picard modular groups PU(2,1,Od)are lattices of the holomorphic isometry group of the complex hyperbolic plane.The class number of the ring of algebraic integers Od in Q(i(?)),is equal to 1 if and only if the Picard modular group has exactly one cusp,and there are only finitely many values of d such that this happens,namely d=1,2,3,7,11,19,43,67,163.We present an effective method to construct coarse fundamental domains for the action of 1-cusped Picard modular groups.As an application,we classify torsion elements up to conjugation in the Picard modular groups PU(2,1,Od)with d=1,2,3,7,11,19,and obtain short presentations for these groups by using a small number of torsion elements as group generators.In addition,we construct neat subgroups of small indexes.