On the invariants of Möbius groupsM(R n )

Abstract

Suppose that $$\operatorname{Re} (a + d^ * ) \in \left\{ {\begin{array}{*{20}c} {( - 2,2),if g(x) is f.p.f. or elliptic,} \\ {\left[ { - 2,2} \right], if g(x) is parabolic,} \\ {( - \infty ,\infty ), if g(x) is loxodromic.} \\ \end{array} } \right.$$ is a Clifford matrix of dimensionn, g(x)=(ax+b)(cx+d) −1. We study the invariant balls and the more careful classifications of the loxodromic and parabolic elements inM(R n ), prove that the loxodromic elements inM(R 2k+1 ) certainly have an invariant ball, expound the geometric meaning of Ahlfors' hyperbolic elements, and introduce the uniformly hyperbolic and parabolic elements and give their identifications. We prove that $$\operatorname{Re} (a + d^ * ) \in \left\{ {\begin{array}{*{20}c} {( - 2,2),if g(x) is f.p.f. or elliptic,} \\ {\left[ { - 2,2} \right], if g(x) is parabolic,} \\ {( - \infty ,\infty ), if g(x) is loxodromic.} \\ \end{array} } \right.$$ These results are fundamental in the higher dimensional Mobius groups, especially in Fuchs groups.