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

Published online by Cambridge University Press:  01 March 2008

SHIGEYASU KAMIYA  and
JOHN R. PARKER
SHIGEYASU KAMIYA
Affiliation:
Okayama University of Science, 1-1 Ridai-cho, Okayama 700-0005, Japan. e-mail: [email protected]
JOHN R. PARKER
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH 1 3LE. e-mail: [email protected]
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 2 , March 2008 , pp. 443 - 455
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Basmajian, A. and Miner, R.. Discrete groups of complex hyperbolic motions. Invent. Math. 131 (1998), 85–136.CrossRefGoogle Scholar
[2]Goldman, W. M.. Complex Hyperbolic Geometry (Oxford University Press, 1999).CrossRefGoogle Scholar
[3]Jiang, Y., Kamiya, S. and Parker, J. R.. Jørgensen's inequality for complex hyperbolic space. Geom. Dedicata. 97 (2003), 55–80.CrossRefGoogle Scholar
[4]Jiang, Y. and Parker, J. R.. Uniform discreteness and Heisenberg screw motions. Math. Zeit. 243 (2003), 653–669.CrossRefGoogle Scholar
[5]Kamiya, S.. Notes on non–discrete subgroups of Û(1,n;). Hiroshima Math. J. 13 (1983), 501–506.Google Scholar
[6]Kamiya, S.. Notes on elements of U(1,n;ℂ). Hiroshima Math. J. 21 (1991), 23–45.Google Scholar
[7]Kamiya, S.. On discrete subgroups of PU(1,2;ℂ) with Heisenberg translations. J. London Math. Soc. 62 (2000), 827–842.CrossRefGoogle Scholar
[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), 689–695.Google Scholar
[9]Ohtake, H.. On discontinuous subgroups with parabolic transformations of the Möbius groups. J. Math. Kyoto Univ. 25 (1985), 807–816.Google Scholar
[10]Parker, J. R.. Shimizu's lemma for complex hyperbolic space. International J. Math. 3 (1992), 291–308.CrossRefGoogle Scholar
[11]Parker, J. R.. Uniform discreteness and Heisenberg translations. Math. Zeit. 225 (1997), 485–505.CrossRefGoogle Scholar
[12]Parker, J. R.. On the stable basin theorem. Canad. Math. Bull. 47 (2004), 439–444.CrossRefGoogle Scholar
[13]Shimizu, H.. On discontinuous subgroups operating on the product of the upper half planes. Ann. of Math. 77 (1963), 33–71.CrossRefGoogle Scholar
[14]Waterman, P. L.. Möbius transformations in all dimensions. Adv. Math. 101 (1993), 87–113.CrossRefGoogle Scholar