Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T12:52:44.900Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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