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The chordal norm of discrete Möbius groups in several dimensions. (English) Zbl 0854.30032

Summary: Let \(d(f,g) = \sup \{d (f(z), g(z)) : z \in \overline \mathbb{C}\}\) where \(f,g\) are Möbius transformations and \(d(z_1, z_2)\) denotes the chordal distance between \(z_1\), \(z_2\) in \(\overline \mathbb{C}\). We show that if \(\langle f,g \rangle\) is a discrete group and if \(fg \neq gf\), then \(\max \{d (f, \text{id}), d(g, \text{id})\} \geq c\) where \(.863 \leq c \leq.911 \cdots\). We also obtain some higher dimensional analogues by means of Clifford numbers.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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