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Commensurators of finitely generated nonfree Kleinian groups. (English) Zbl 1237.20044
This article was inspired by the question of whether there is a broad generalization of the theorem of Margulis by replacing the finite covolume hypothesis by the weaker assumption that the group only be Zariski dense. Let \(\Gamma\) be a finitely generated torsion-free Kleinian group of the first kind which is not a lattice and \(C(\Gamma)\) be the commensurator of \(\Gamma\). The authors show that if \(\Gamma\) is not free and contains no parabolic elements then \(C(\Gamma)\) is discrete, furthermore, \([C(\Gamma ):\Gamma]=\infty\) if and only if \(\Gamma\) is a fiber group, in this case, \(C(\Gamma)\) is a lattice.

MSC:
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
22E40 Discrete subgroups of Lie groups
57M50 General geometric structures on low-dimensional manifolds
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