Picard Modular Groups And Spherical CR Geometry

As the generalization of classic modular group PSL(2, Z) in high dimensionalcomplex hyperbolic space, Picard modular groups PU(2,1; Od) are the simplestcomplex hyperbolic arithmetic lattices, where Odis the ring of algebraic integersin the feld Q(id) and d is a square-free positive integer. Since studying thePicard modular groups will supply very useful examples for the research of thegeneral theory of complex hyperbolic discrete groups, it is an important subjectin the theory of complex hyperbolic geometry.In this thesis we frstly studied the generators of the Picard modular groupsPU(2,1; Od)(d=2,3,7,11). We obtained a new system of fnite generators ofPU(2,1; O3) by using the method of continued fraction algorithm. For the others,we frstly found a new fnite generator system of the stabilizer subgroup fxing thepoint at infnity, and then with Zhao’s results we obtained a new system of fnitegenerators of PU(2,1; Od)（d=2,7,11）. About the high dimensional Picardmodular groups, we found a system of fnite generators of the three dimensionalPicard modular group PU(3,1; O3) by using the continued fraction algorithm.According to Serre’s theorem, we call a group has Property (FA) if it is fnitelygenerated and does not split into nontrivial free product with amalgamation anddoes not admit any homomorphism onto Z. Wether a group has Property (FA)is an important problem in the feld of semisimple Lie groups.Secondly we studied the Property (FA) of the Gauss-Picard modular groupPU(2,1; O1) and the sister of the Eisenstein-Picard modular group PU(2,1; O3).Eisenstein-Picard modular group and its sister group are the only two fundamentalgroups of the arithmetic, cusped, complex hyperbolic orbifolds of minimal volume.In2007, Stover proved that Eisenstein-Picard modular group PU(2,1; O3), andhe asked wether the other Picard modular groups have Property (FA)? We provedthat the Gauss-Picard modular group has Property (FA), which is a partiallypositive answer for Stover’s question. We proved the sister of the Eisenstein-Picard modular group has Property (FA). It means that the Eisenstein-Picardmodular group and the sister of the Eisenstein-Picard modular group, as the onlytwo arithmetic fundamental groups of cusped, complex hyperbolic orbifolds ofminimal volume, have the same property of Property (FA).In the theme of geometry, an important problem is to study the relationshipsof topology properties and geometric properties under a fne geometry structure of a space. A spherical CR structure on a3-dimensional manifold is a maximalsystem of coordinate charts in S3, and the transformation for the same domainis the restriction of an element in the group PU(2,1). It is well known that allof the3-dimensional manifolds can equipped with contact structure and naturallythe CR structure. This fact suspect that spherical CR structure should be a basicproblem in the study of3-dimensional manifolds, therefore, a natural question isto ask when a3-dimensional manifold has a spherical CR structure.Lastly, we researched the spherical structure on the complement of fgure eightknot M=S3K, which is a3-dimensional hyperbolic manifold. Up to conju-gation, Falbel constructed three diferent representations ρi(i=1,2,3) from thefundamental group of the complement of fgure eight knot π1(M) to PU(2,1) bythe method of gluing two tetrahedra. He studied the frst representation ρ1moreprecisely, proved that the frst representation is discrete and there is a branchedspherical CR structure, more interesting, the limit set of the group ρ1(π1(M)) isthe whole boundary S3of the complex hyperbolic space. In this paper, we s-tudied the second representation ρ2and proved that it is discrete and the groupρ2(π1(M)) contained in the Picard modular group PU(2,1; O7) as an infnity in-dex subgroup, and there is a branched spherical CR structure associated to thisrepresentation. Besides, we compared the commonality and diferences of the threerepresentations. It is easy to see that the biggest commonality is that all of thegroups ρi(π1(M))(i=1,2,3) are arithmetic, since ρ1(π1(M)) is contained in theEisenstein-Picard modular group PU(2,1; O3) as an infnity index subgroup andρ3(π1(M)) is conjugated to ρ2(π1(M)) which means that it is also a subgroup ofinfnity index in the Picard modular group PU(2,1; O7). The biggest diferenceis that the ρ1(π1(M)) contains a surface subgroup ρ1(F2) of infnity index (F2isthe fundamental group of the once punctured torus since the complement of thefgure eight knot is the once punctured torus bundle over a circle), and ρi(F2) is afnite index (index three) subgroup ρi(π1(M)) when i=2,3, which means that thesecond and third representations are almost determined by the surface subgroupcontained in the fundamental group of the manifold.