A Number Of Problems On Lie Group Sp(n,1)

Hyperbolic manifolds and discrete groups are important research directions in the geomet-ric theory of modern complex analysis and it is very novel to study the hyperbolic geometry by the theory of Lie group.Adeboye and Wei obtained the lower bound for the volumes of real and complex hyperbolic orbifolds in[1]and[3].This thesis focuses on the Lie group Sp?n,1?.Lie group has a group structure,as well as be-ing a differential manifold which has properties on analysis and geometry.Analogue to Adeboye and Wei's approach in complex hyperbolic space,we obtain the lower bound for the volumes of quaternionic hyperbolic spaces;at the same time we reestimate the result of Adeboye and Wei in complex case.Roughly speaking,our approch is as follows:We construct a Riemannian submersion from the quotient Sp?n,1?/?to the quotient H_Hⁿ/?.With this Riemannian submer-sion,we can employ Wang's result[21,Theorem 5.2]to produce an inscribed ball of radius₂^RSp?n,1?in H_Hⁿ/?and obtain the lower bound for the volumes of inscribed ball by a comparison theorem of Gunther[9,Theorem 3.101].In Chapter 1,we mainly introduce the background,research status and significance of study the lower bound of on volumes of quaternionic hyperbolic n-orbifolds.In Chapter 2,preliminary knowledge are introduced,including quaternionic and quater-nionic hyperbolic space,Lie algebra and Killing form of Sp?n,1?.In Chapter 3,we obtain the real matrix representation of the adjoint action of Lie algebra sp?1,1?,also find out the real dimension of Lie group Sp?1,1?,Lie basis,Lie bracket,Cartan decomposition of sp?1,1?.In Chapter 4,we obtain the lower bound of volume of a quaternionic hyperbolic n-orb-ifold.In this process,we study a number of Lie algebra problems of sp?n,1?,including the real dimension,Lie basis,Lie bracket,Cartan decomposition,canonical metric of Sp?n,1?.At the same time,we introduce the Riemannian submersion to the Lie group of Sp?n,1?,and com-bine with some Lie algebra problems to find out the section curvature of Sp?n,1?,and combine Wang^?s result[21,Theorem 5.2]and a comparison theorem of Gunther[9,Theorem 3.101]to obtain the lower bound of volume of a quaternionic hyperbolic n-orbifold.In Chapter 5,using the method of the fourth chapter,we reestimate the lower bound of n-orbifold volume of complex hyperbolic space.