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Recognizing constant curvature discrete groups in dimension 3. (English) Zbl 0910.20024
The goal of the paper is to characterize those groups \(G\) which can act properly discontinuously, isometrically and cocompactly on hyperbolic 3-space \(\mathbb{H}^3\) (that is as cocompact Kleinian groups), in terms of the combinatorics of the action of \(G\) on its space at infinity. The Cayley graph \(\Gamma\) of such a group \(G\) has negative curvature (the group is word hyperbolic in the sense of Gromov), and the space at infinity \(\partial\Gamma\) of the Cayley graph is the 2-sphere \(S^2\). The main result of the paper states that these two conditions, together with a third more technical but crucial condition that certain disks cover \(\partial\Gamma=S^2\) in a combinatorially nice way, are necessary and sufficient for the existence of an action of \(G\) as a cocompact Kleinian group. One of the ingredients of the proof, apart from the theory of word hyperbolic groups, is the Sullivan-Tukia theorem that a group \(G\) acts geometrically on \(\mathbb{H}^3\) if and only if \(G\) acts discretely and uniformly quasiconformally on the 2-sphere. One of the principal difficulties comes from the problem how to distinguish a group acting on a space of constant negative curvature from one acting on a space of variable negative curvature (in higher dimensions these classes of groups are different, in dimension three Thurston’s geometrization conjecture would imply that they coincide, as they do in dimension two).

MSC:
20F65 Geometric group theory
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M07 Topological methods in group theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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