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


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]Basmajian, A. and Miner, R.. Discrete groups of complex hyperbolic motions. Invent. Math. 131 (1998), 85136.
[2]Goldman, W. M.. Complex Hyperbolic Geometry (Oxford University Press, 1999).
[3]Jiang, Y., Kamiya, S. and Parker, J. R.. Jørgensen's inequality for complex hyperbolic space. Geom. Dedicata. 97 (2003), 5580.
[4]Jiang, Y. and Parker, J. R.. Uniform discreteness and Heisenberg screw motions. Math. Zeit. 243 (2003), 653669.
[5]Kamiya, S.. Notes on non–discrete subgroups of Û(1,n;). Hiroshima Math. J. 13 (1983), 501506.
[6]Kamiya, S.. Notes on elements of U(1,n;ℂ). Hiroshima Math. J. 21 (1991), 2345.
[7]Kamiya, S.. On discrete subgroups of PU(1,2;ℂ) with Heisenberg translations. J. London Math. Soc. 62 (2000), 827842.
[8]Kamiya, S. and Parker, J. R.. On discrete subgroups of PU(1,2;ℂ) with Heisenberg translations II. Rev. Roumaine Math. Pures Appl. 47 (2002), 689695.
[9]Ohtake, H.. On discontinuous subgroups with parabolic transformations of the Möbius groups. J. Math. Kyoto Univ. 25 (1985), 807816.
[10]Parker, J. R.. Shimizu's lemma for complex hyperbolic space. International J. Math. 3 (1992), 291308.
[11]Parker, J. R.. Uniform discreteness and Heisenberg translations. Math. Zeit. 225 (1997), 485505.
[12]Parker, J. R.. On the stable basin theorem. Canad. Math. Bull. 47 (2004), 439444.
[13]Shimizu, H.. On discontinuous subgroups operating on the product of the upper half planes. Ann. of Math. 77 (1963), 3371.
[14]Waterman, P. L.. Möbius transformations in all dimensions. Adv. Math. 101 (1993), 87113.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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