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On the Margulis constant for Kleinian groups. I. (English) Zbl 0854.30031

Summary: The Margulis constant for Kleinian groups is the smallest constant \(c\) such that for each discrete group \(G\) and each point \(x\) in the upper half space \(\mathbb{H}^3\), the group generated by the elements in \(G\) which move \(x\) less than distance \(c\) is elementary. We take a first step towards determining this constant by proving that if \(\langle f,g \rangle\) is nonelementary and discrete with \(f\) parabolic or elliptic of order \(n\geq 3\), then every point \(x\) in \(\mathbb{H}^3\) is moved at least distance \(c\) by \(f\) or \(g\) where \(c = .1829 \dots\). This bound is sharp.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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