On the Invariants of Mobius Groups M(R~n)

Suppose that g=[a b c d] is a Clifford matrix of dimension n, g(x)=(ax+b)(cx+d). We study the invariant balls and the more careful classifications of the loxodromic andparabolic elements in M(R~n), prove that the loxodromic elements in M(R) certainly havean invariant ball, expound the geometric meaning of Ahlfors’ hyperbolic elements, and introducethe uniformly hyperbolic and parabolic elements and give their identifications. We prove that (-2, 2), if g(x) is f.p.f, or elliptic,Re(a+d~*)∈{[-2, 2}, if g(x) is parabolic, (-∞, ∞), if g(x) is loxodromic. These results are fundamental in the higher dimensional Mbius groups, especially in Fuchsgroups.