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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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