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A discreteness criterion for subgroups of \({\mathbf{PU}(2,1)}\) with screw parabolic elements. (English) Zbl 1296.22012

The authors establish a discreteness criterion for subgroups of \(\mathrm{PU}(2,1)\) that contain a screw parabolic element \(g\). The work relies on the geometry of the Margulis region and it is motivated by works of V. Erlandsson and S. Zakeri [“On Margulis cusps of hyperbolic 4-manifolds”, arXiv:1304.5316; “A discreteness criterion for groups containing parabolic isometries”, arXiv:1304.2298]. An explicit region precisely invariant under the stabilizer of the fixed point of \(g\) is given.
The result in this paper is a generalization of a result which gives a necessary condition for a subgroup of \(\mathrm{PSL}(2,\mathbb{R})\) to be discrete and which is due to H. Shimizu [Ann. Math. (2) 77, 33–71 (1963; Zbl 0218.10045)].

MSC:

22E40 Discrete subgroups of Lie groups
51M10 Hyperbolic and elliptic geometries (general) and generalizations

Citations:

Zbl 0218.10045
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References:

[1] Erlandsson, V.; Zakeri, S., On Margulis cusps of hyperbolic 4-manifolds, 34 pages · Zbl 1310.22008
[2] Erlandsson, V.; Zakeri, S., A discreteness criterion for groups containing parabolic isometries, 8 pages · Zbl 1360.22015
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