Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

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


a1 Okayama University of Science, 1-1 Ridai-cho, Okayama 700-0005, Japan. e-mail:

a2 Department of Mathematical Sciences, University of Durham, Durham DH 1 3LE. e-mail:


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.

(Received February 03 2006)

(Revised May 10 2007)

(Online publication February 07 2008)