Geodesics In Hyperbolic 3-Space

This paper begins with studying the elementary of two-generator group. At first it gives the connection between elementary of one group and that of its subgroup and obtains:if and only if for any two elements f and g in G, is elementary, G is elementary. This result simplifies the judgment about the elementary of one general group. This paper analyzes the situation of the elementary of two-generator group, when one generator is loxodromic and the other is a general Mobius transformation.Then this paper studies the discrete of two-generator group, obtains two deformations about the J(?)rgensen's inequality, on the basis of the relationships between the discrete of one group and that of its subgroup by matlab.Discussing whether one group is discrete and elementary lays a strong foundation for studying the geodesics in hyperbolic 3-space.As for the problem of geodesics in hyperbolic 3-space, this paper begins with studying the axes of Mobius transformations, obtains two main results. One is about the relationship between two generators'order and the angle between their axes in two-generator group. Furthermore, this paper gives the special connection when the angle is a right angle. The other is about the relationship between two generators' translation length and the distance of their axes, when both the two generators are loxodromics and their axes coplanar but non-intersecting.At present, the focal point of the research about the geodesic is on the collar in complex hyperbolic manifolds. On the basis of seniors'plentiful fruits, this paper obtains the relationship between the multiplier of a loxodromic element and the existence of a precisely invariant tube about the axis. This will be a complement to the results of Markham. S and Parker J. R. Then this paper obtains the volume of embedded collar about a geodesic in complex hyperbolic manifolds.In the end,this paper give the lower bound of the distance between two sufficient short, simple, non-intersecting and closed geodesics in complex hyperbolic manifolds.