The Isolation Of Transformations In Two-generater Subgroups Of Discrete M(?)bius Groups

The subject matter of the geometry of discrete groups has now been stud-ied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds. This paper is a quantitative statement about the isoland nature of I within a discrete non-elementary Mobius group, and contains two part:geometric and algebraic.(Ⅰ) Geometrically, it is concerned with the relationship between fixed points and hypercyclics or horoballs of Mobius transformations in discrete groups. The analogous study is done by A. F. Bearton. He characterized horoballs and horospheres in terms of the geometry of H3 alone, as a geomet-ric interpretation of J?rgensen's inequality. The main research achievements are as follows:Ifis discrete and g,f have the same fixed points, then the one which has larger norm has smaller hypercyclic when g, f are parabolic; the one which has larger trace has smaller horoball when g, f are hyperbolic; the one which has larger rotation angle has smaller horoball when g, f are elliptic. Ifis discrete and g, f have no fixed point, and then f is conjugate to g, then there exists a positive number such that hypercyclics or horoballs of g, f have no intersection. These results is given considering hyperbolic metric.(Ⅱ) Algebraically, it is concerned with Loxodromic Mobius Transforma-tions with Coplanar Axes in a Kleinian Group. In this field F. W. Gehring and G. J. Martin have done lots of important researches. They gave some equalities and inequalities about trace, and the collar a(n) of two elliptic transformations in a discrete non-elementary group. In this section, the main results are about the distance between their axes and some estimates about their translation lengths. The research achievements are as follows:Ifis a Kleinian group, if f and g are hyperbolic, and if ax(f) and ax(g) are coplanar but do not intersect,δis the distance between ax(f) and ax(g), thenifis discrete, if f and g are loxodromics, if ax(f) and ax(g) are coplanar but do not intersect, and if sinh(δ)≥1, then |β(f)β(g)|sinh4/3(δ)≥b> 0.792..., andif is discrete, if f is loxodromic and g is elliptic of order n≥3, andif ax(f) and ax(g) are coplanar but do not intersect, then sinh(t(f))sin2(π/n)sinh2(δ)≥0.333....These inequalities characterizes the isoland nature of I within a discrete group, like J(?)rgensen's inequality.