Möbius transformations in several dimensions. (English) Zbl 0793.15019

Waterman, P. L.

Adv. Math. 101, No. 1, 87-113 (1993).

From the author’s introduction: L. V. Ahlfors [Differential Geometry and Complex Analysis, 65-73 (1985; Zbl 0569.30040)] shows how a \(2\times 2\)-matrix with entries in a Clifford algebra may be used to describe a Möbius transformation of \(\mathbb{R}^ n\cup\{\infty\}\).
We give a different development of Clifford matrices and discuss their relationship to hyperbolic isometries. A discreteness condition generalizing Jørgensen’s inequality is then obtained and utilized to describe the nature of parabolic fixed points in higher dimensions. Although they cannot be conical limit points, we give an example to show that they may be horospherical limit points.

Reviewer: A.Helversen-Pasotto (Nice)

MSC:

15A66	Clifford algebras, spinors
53C56	Other complex differential geometry

Keywords:

Clifford algebra; Möbius transformation; Clifford matrices; hyperbolic isometries; Jørgensen’s inequality; parabolic fixed points; horospherical limit points