Some Properties Of Schottky Groups Of Genus Two

The main purpose of this thesis is to investigate the properties of Schottky groupsof genus two.Schottky groups is a class of important Mo¨bius groups, and also, it is a class ofspecial Klein groups. As for the properties of Schottky groups of genus two, this thesisstudies its properties from two aspects. On one hand, according to the value of t₁,t₂andρ, we divide the groups into eight classes firstly, and then work out the di?erentforms of Jφrgensen's numbers of the third, the sixth and the eighth. We get to deducethe theorems of equivalence according to the characteristics of these three classes. Onthe other hand, we can deduce the relation of di?erent Schottky groups by using thedefinition of Nielsen inverse transformation and study the hyperbolic transformationsof the third, the sixth and the eighth.The thesis has five parts.Chapter 1 provides the background information , significance and progress aboutSchottky groups of genus two and introduces the organization of the whole thesis.Chapter 2 introduces the basic concepts and properties of Schottky groups ofgenus two, including the definitions of Schottky groups, Mo¨bius groups and ect. Andthese are used for the latter.Chapter 3 investigates the relationships among t₁,t₂ andρin Nielsen inversetransformation. Furthermore, it discovers that the mapping Nj(?) 1and the mapping Njhave the same function on t1,t1 andρ, where j = 1,2.Chapter 4 introduces Jφrgensen's numbers of Schottky groups of three types andpresents the concrete forms.Chapter 5 studies the relationships among three classes of Schottky groups ofgenus two under Nielsen inverse transformation and finds out the properties of them.