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On discreteness of commensurators. (English) Zbl 1209.57010
Motivated by Margulis’ celebrated characterization of arithmeticity of irreducible lattices in semisimple Lie groups in terms of density of the commensurator, the paper concerns the problem to describe the commensurators of Zariski dense subgroups of semisimple Lie groups. It starts with the observation that for Zariski dense subgroups of semisimple Lie groups occuring as isometry groups of rank one symmetric spaces, commensurators are discrete provided that the domain of discontinuity is nonempty, and then presents a generalization also for arbitrary symmetric spaces.
Then, shifting the interest to Kleinian groups, it is proved that for all finitely generated, Zariski dense, infinite covolume discrete subgroups \(G\) of Isom(\(\mathbb H^3\)), commensurators are discrete; more generally, [Comm(\(G):G] < \infty\) unless \(G\) is virtually a fiber subgroup, in which case Comm(\(G\)) is the fundamental group of a virtually fibered finite volume hyperbolic 3-manifold. As the author notes, together these results prove discreteness of commensurators for all known examples of finitely presented, Zariski dense, infinite covolume discrete subgroups of Isom(\(X\)) for \(X\) a symmetric space of noncompact type.

MSC:
57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20F67 Hyperbolic groups and nonpositively curved groups
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