Discreteness Of Isometry Group In Complex Hyperbolic Space

Hyperbolic geometry keeps closely relation with Teichm¨uller Space, complex dynamic system, low dimensional topolofy, hyperbolic manifold, etc. Complex hyperbolic geometry, quaternion hyperbolic geometry are generalizations of hyperbolic geometry. Picard, Giraud, Cartan, Chen, Greenberg, etc., have established a foundation of complex hyperbolic geometry. After that, Goldman, Schwartz,Parker, etc. have obtained remarkable accomplishments and enriched greatly the theory of complex hyperbolic geometry, and stimulated a flourishing interesting of researches to study further complex hyperbolic geometry.Our main work is to study discrete criteria of isometry groups and conjugation of two-generator subgroups in complex hyperbolic space, J?rgensen equality and algebraic convergence of isometry group in quaternion hyperbolic space. The paper is organized as follows:In the first chapter, the research background, significance and main innovation points are introduced.In the second chapter, the basic knowledge about complex hyperbolic geometry are reviewed.In the third chapter, three main problems are discussed. The first problem is the conjugation of two-boundary-elliptic-generator subgroups. The second problem is discreteness of two-generator subgroups. The third problem is the relations between discreteness and properly discontinuity, discreteness and limit sets.In the fourth chapter, utilizing test maps to judge the discreteness of isometry groups in complex hyperbolic space, the discreteness criteria for subgroups of P U(n, 1) are obtained.In the fifth chapter, J?rgensen equalities of quaternion hyperbolic space are studied, and algebraic convergence criterion of isometry groups in quaternion hyperbolic space is obtained.