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Conjugacy invariants of Möbius transformations. (English) Zbl 0721.30035
This paper is devoted to generalizing the trace-squared (conjugation) invariant of Möbius transformations with complex entries to elements of a certain set of Clifford matrices ${\cal C}\sp n$. These were first considered by {\it K. Th. Vahlen} [Math. Ann. 55, 585-593 (1902)] and later by {\it L. V. Ahlfors} [see e.g. Ann. Acad. Sci. Fenn., Ser. A I 105, 15-27 (1985; Zbl 0586.30045)]. The Clifford matrices considered here are slightly different from those obtained by Ahlfors and provide a generalization to orientation-reversing maps. The main theorem is that, for each $k=0,1,2,...,n+2$, the $T\sb k$ function (the definition is too complicated to give here) $$ T\sb k: M(2,{\cal C}\sp n)\to {\bbfR} $$ is a conjugation invariant. Here ${\cal C}\sp n$ is the algebra of Clifford numbers generated by n roots of -1. The conjugation is by Clifford matrices. Among the properties of the invariant $T\sb k$ are the following: (i) $T\sb k(-g)=T(g)$ for all $g\in {\cal C}\sp n$. Thus $T\sb k$ is well-defined on the projectivization of ${\cal C}\sp n.$ (ii) $T\sb k(g)=0$ for all k odd (resp. even) if g is orientation preserving (resp. reversing) $$ \sum\sp{n+2}\sb{0}T\sb k(g)=0. $$ (iii) $g\in {\cal C}\sp n$ is hyperbolic if and only if $T\sb k(g)<0$ for the largest k so that $T\sb k(g)\ne 0.$ (iv) $g\in {\cal C}\sp n$ is elliptic if and only if the quadratic form $N(z):=z\bar z$, when restricted to the kernel of Id-r(g) is not positive semidefinite. Here r takes g from the isometries of the ball model of hyperbolic space to the hyperboloid model.
Reviewer: W.Abikoff (Storrs)

30G35Functions of hypercomplex variables and generalized variables