On The Proper Conjugation For Complex Hyberbolic Discrete Groups

The hyperbolic geometry of the non-Euclidean geometry is a very important re-search direction of the modern complex analysis in geometry theory. The achievementin research has important applications in some aspects such as Riemann curved sur-face, lower-dimensional topology, dynamical systems and Teichmu¨ller space. Researchon the real hyperbolic geometry reached its peak in the20th century. An increasingnumber of experts and scholars were interested in complex hyperbolic and complexhyperbolic group with perfection of real hyperbolic theory. So, many famous achieve-ments were put forward. In the recent years, many experts and scholars paid attentionto the Co-Hopf problem that research the arbitrary one in group G to itself withinjective homomorphism. The proper conjugation of isometric group in complex hy-perbolic is a special case of Co-Hopf problem. In this paper, we mainly discuss properconjugation of diverging complex hyperbolic discrete group by adopting measure ofPatterson-Sullivan.The main content contain three chapters as follows:The first chapter which is introduction mainly introduces backgrounds and achieve-ments. It also has given the innovation of this paper and the main work.The second chapter presents relevant pre-knowledge of complex hyperbolic discretegroup such as the first Hermitian form, the second Hermitian form, the model ofcomplex hyperbolic space and the classification and limit set of isometric conversion.In the third chapter, the proper conjugation of diverging complex hyperbolic dis-crete group has been researched. Poincare series in the complex hyperbolic space isgiven at first. Conformal density and measure of Patterson-Sullivan are introducedthen. The invariance of measure of Patterson-Sullivan in the complex hyperbolic spaceis demonstrated and the proper conjugation of complex hyperbolic discrete group isconstructed next. At last, we proved complex hyperbolic discrete group without properconjugation by utilizing uniqueness of measure of Patterson-Sullivan and discretenessof proper conjugation of isometric group in complex hyperbolic space. Thus, The realhyperbolic space results extend the solution to complex hyperbolic space.