M(?)bius Groups In Finite And Infinite Dimensions

Mobius group theory has applications in many fields, such as complex analytic dynamics, Teichmiiller spaces and Sobolev spaces, potential theory, physics, engineering technology and so on. There are remarkable connections among discrete Mobius groups, Riemann surfaces and hyperbolic manifolds. The theory of infinite dimensional Banach manifolds is an indispensible part of that of manifolds. It is valuable to generalize Mobius groups to infinite dimensional inner product spaces.In Section 1, let G be a non-elementary Mobius group acting on the extended plane and g0 ∈ G a loxodromic type Mobius transformations. We show that G is discrete if and only if for each g G G the group < g,g0 > is discrete. Moreover, the test element g0 even need not be in G. This improves the well-known discreteness criteria established by J0rgensen.In Section 2, we consider Mobius groups of high dimensions. G. J. Martin and Fang A.N., Nai B. established an important convergence theorem in all dimensions but added the restrictions that involved groups are either finitely generated or satisfy Condition A, respectively. We find a new approach to show that these restrictions are not necessary.In Section 3, we study Mobius groups acting on infinite-dimensional inner product spaces. First, we establish equivalences between Mobius maps and sphere-preserving bijections. We also give a new characteristic of Mobius maps by using Apollononius k-complex and prove the conformality of Mobius maps. It is known that a Mobius transformations of Rn is a finite composition of reflections in spheres. Here we give an example to show that this phenonmenon does not hold in the infinite case. In another development, denote by M(B) the groups of Mobius maps which preserves the unit ball B = {x ∈ E||x| < 1}. After introducing a hyperbolic metric on B, we prove that M(B) is exactly the isometric group with respect to this metric. F.W. Gehring and G.J. Martin introduced the notation of convergence groups and proved Mobius groups in Rn are convergence groups. Our example shows infinite dimensional Mobius groups do not have this topological property.