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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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