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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 144, Issue 2
  • March 2008, pp. 443-455

Discrete subgroups of PU(2, 1) with screw parabolic elements

  • DOI:
  • Published online: 01 March 2008

We give a version of Shimizu's lemma for groups of complex hyperbolic isometries one of whose generators is a parabolic screw motion. Suppose that G is a discrete group containing a parabolic screw motion A and let B be any element of G not fixing the fixed point of A. Our result gives a bound on the radius of the isometric spheres of B and B−1 in terms of the translation lengths of A at their centres. We use this result to give a sub-horospherical region precisely invariant under the stabiliser of the fixed point of A in G.

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[1]A. Basmajian and R. Miner . Discrete groups of complex hyperbolic motions. Invent. Math. 131 (1998), 85136.

[3]Y. Jiang , S. Kamiya and J. R. Parker . Jørgensen's inequality for complex hyperbolic space. Geom. Dedicata. 97 (2003), 5580.

[4]Y. Jiang and J. R. Parker . Uniform discreteness and Heisenberg screw motions. Math. Zeit. 243 (2003), 653669.

[7]S. Kamiya . On discrete subgroups of PU(1,2;ℂ) with Heisenberg translations. J. London Math. Soc. 62 (2000), 827842.

[10]J. R. Parker . Shimizu's lemma for complex hyperbolic space. International J. Math. 3 (1992), 291308.

[11]J. R. Parker . Uniform discreteness and Heisenberg translations. Math. Zeit. 225 (1997), 485505.

[12]J. R. Parker . On the stable basin theorem. Canad. Math. Bull. 47 (2004), 439444.

[13]H. Shimizu . On discontinuous subgroups operating on the product of the upper half planes. Ann. of Math. 77 (1963), 3371.

[14]P. L. Waterman . Möbius transformations in all dimensions. Adv. Math. 101 (1993), 87113.

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  • EISSN: 1469-8064
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