Skip to main content
    • Aa
    • Aa

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] A. Basmajian and R. Miner . Discrete groups of complex hyperbolic motions. Invent. Math. 131 (1998), 85136.

[3] Y. Jiang , S. Kamiya and J. R. Parker . Jørgensen's inequality for complex hyperbolic space. Geom. Dedicata. 97 (2003), 5580.

[4] Y. Jiang and J. R. Parker . Uniform discreteness and Heisenberg screw motions. Math. Zeit. 243 (2003), 653669.

[7] S. Kamiya . On discrete subgroups of PU(1,2;ℂ) with Heisenberg translations. J. London Math. Soc. 62 (2000), 827842.

[10] J. R. Parker . Shimizu's lemma for complex hyperbolic space. International J. Math. 3 (1992), 291308.

[11] J. R. Parker . Uniform discreteness and Heisenberg translations. Math. Zeit. 225 (1997), 485505.

[12] J. R. Parker . On the stable basin theorem. Canad. Math. Bull. 47 (2004), 439444.

[13] H. Shimizu . On discontinuous subgroups operating on the product of the upper half planes. Ann. of Math. 77 (1963), 3371.

[14] P. L. Waterman . Möbius transformations in all dimensions. Adv. Math. 101 (1993), 87113.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 77 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 25th May 2017. This data will be updated every 24 hours.