The Isometry Groups In Inner Product Spaces And The Discrete M(?)bius Groups In Higher Dimensions

In this paper, we mainly study the isometry groups in infinite dimensions innerproduct spaces and the discrete Mo|¨bius groups in higher dimensions. this thesis isarranged as follows:In chapter 1, we provide some background of our problems which will be intro-duced and the statement of our main results.In chapter 2, we discuss the isometry groups consists of Mo|¨bius transformationswhich mapping the unit ball B onto B in inner product spaces: First, we investigatethe relationship between the re?ections and the Mo|¨bius transformations, we also gainone necessary and su?cient condition for a map to be a Mo|¨bius transformations ininner product spaces, thus we have some conclusions like in Euclidean spaces; Next, wegive the precise description about equicontinuity on Mo|¨bius transformations in innerproduct spaces. Last, we establish the Jφgensen inequality in special case by usingpure algebraic method.In chapter 3, we discuss the discrete Mo|¨bius groups in higher dimensions, by us-ing the chordal norm and Cli?ord matrices we obtain three necessary condition of thediscrete groups which are generated by Mo|¨bius transformations in higher dimensionsEuclidean spaces .