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DISCRETENESS CRITERIA FOR ISOMETRIC GROUPS ACTING ON COMPLEX HYPERBOLIC SPACES

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  17 March 2010

XI FU
XI FU*
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, PR China (email: [email protected])
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Abstract

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In this paper, four new discreteness criteria for isometric groups on complex hyperbolic spaces are proved, one of which shows that the Condition C hypothesis in Cao [‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc.78 (2008), 211–224, Theorem 1.4] is removable; another shows that the parabolic condition hypothesis in Li and Wang [‘Discreteness criteria for Möbius groups acting on II’, Bull. Aust. Math. Soc.80 (2009), 275–290, Theorem 3.1] is not necessary.

Keywords

discreteness criterianonelementarycomplex hyperbolic spacesloxodromic

MSC classification

Secondary: 20F67: Hyperbolic groups and nonpositively curved groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 3 , June 2010 , pp. 481 - 487
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Basmajian, A. and Miner, R., ‘Discrete groups of complex hyperbolic motion’, Invent. Math. 131 (1998), 85–136.CrossRefGoogle Scholar
[2]Cao, W., ‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc. 78 (2008), 211–224.CrossRefGoogle Scholar
[3]Cao, W. and Wang, X., ‘Discreteness criteria and algebraic convergence theorem for subgroups in PU (1,n;ℂ)’, Proc. Japan Acad. 82 (2006), 49–52.Google Scholar
[4]Chen, S. S. and Greenberg, L., Hyperbolic Spaces: Constributions to Analysis (Academic Press, New York, 1974), pp. 49–87.Google Scholar
[5]Dai, B., Fang, A. and Nai, B., ‘Discreteness criteria for subgroups in complex hyperbolic space’, Proc. Japan Acad. 77 (2001), 168–172.Google Scholar
[6]Goldman, W. M., Complex Hyperbolic Geometry (Oxford University Press, New York, 1999).CrossRefGoogle Scholar
[7]Jørgensen, T., ‘On discrete groups of Möbius transformation’, Amer. J. Math. 98 (1976), 739–749.CrossRefGoogle Scholar
[8]Kamiya, S., ‘Notes on elements of U(1,n:ℂ)’, Hiroshima Math. J. 21 (1991), 23–45.Google Scholar
[9]Kamiya, S., ‘Chordal and matrix norms of unitary transformations’, First Korean–Japanese Colloquium on Finite or Infinite Dimensional Complex Analysis (1993), 121–125.Google Scholar
[10]Li, L. and Wang, X., ‘Discreteness criteria for Möbius groups acting on II’, Bull. Aust. Math. Soc. 80 (2009), 275–290.CrossRefGoogle Scholar
[11]Parker, J. R., ‘Uniform discreteness and Heisenberg transformation’, Math. Z. 225 (1997), 485–505.CrossRefGoogle Scholar
[12]Tukia, P. and Wang, X., ‘Discreteness of subgroup of SL(2,ℂ) containing elliptic elements’, Math. Scand. 91 (2004), 214–220.CrossRefGoogle Scholar
[13]Wang, X., ‘Dense subgroups of n-dimensional Möbius groups’, Math. Z. 243 (2003), 643–651.CrossRefGoogle Scholar
[14]Wang, X., Li, L. and Cao, W., ‘Discreteness criteria for Möbius groups acting on ’, Israel J. Math. 150 (2005), 357–368.CrossRefGoogle Scholar
[15]Yang, S., ‘On the discreteness in SL(2,ℂ)’, Math. Z. 255 (2007), 227–230.CrossRefGoogle Scholar
[16]Yang, S., ‘Discreteness criteria for isometric groups of real and complex hyperbolic space’, Indiana Univ. Math. J. 58 (2009), 1443–1456.CrossRefGoogle Scholar