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

<正> Suppose that g=[a b c d] is a Clifford matrix of dimension n, g（x）=（ax+b）(cx+d)-1. 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(R2k+1) 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.