Osaka Journal of Mathematics

Decomposition of complex hyperbolic isometries by two complex symmetries

Xue-Jing Ren, Bao-Hua Xie, and Yue-Ping Jiang

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Let $\mathbf{PU}(2,1)$ denote the holomorphic isometry group of the $2$-dimensional complex hyperbolic space $\mathbf{H}_{\mathbb{C}}^{2}$, and the group $\mathbf{SU}(2,1)$ is a 3-fold covering of $\mathbf{PU}(2,1)$: $\mathbf{PU}(2,1)=\mathbf{SU}(2,1)/\{\omega I:\omega^{3}=1\}$. We study how to decompose a given pair of isometries $(A,B)\in \mathbf{SU}(2,1)^{2}$ under the form $A=I_{1}I_{2}$ and $B=I_{3}I_{2},$ where the $I_{k}$'s are complex symmetries about complex lines. If $(A,B)$ can be written as above, we call it is $\mathbb{C}$-decomposable. The main results are decomposability criteria, which improve and supplement the result of [17].

Article information

Osaka J. Math. Volume 54, Number 4 (2017), 661-677.

First available in Project Euclid: 20 October 2017

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Zentralblatt MATH identifier

Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Secondary: 20E45: Conjugacy classes


Ren, Xue-Jing; Xie, Bao-Hua; Jiang, Yue-Ping. Decomposition of complex hyperbolic isometries by two complex symmetries. Osaka J. Math. 54 (2017), no. 4, 661--677.

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  • S. Chen and L. Greenberg: Hyperbolic spaces, in Contributions to Analysis. Academic Press, New York (1974), 49–87.
  • D.Z. Djokovic: The product of two involutions in the unitary group of a hermitian form, Indiana Univ. Math. J. 21 (1971), 449–456.
  • E.W. Ellers: Cyclic decomposition of unitary spaces, J. Geom. 21 (1983), 101–107.
  • E. Falbel and V. Zocca: A Poincar$\acute{e}$'s polyhedron theorem for complex hyperbolic geometry, J. Reine Angew. Math. 516 (1999), 133–158.
  • J. Gilman: A discreteness condition for subgroups of $\mathbf{PSL}(2,C)$, Contemp. Math. 211 (1997), 261–267.
  • W.M. Goldman and J. Parker: Complex hyperbolic ideal triangle groups, Journal für die reine und angewandte Math. 425 (1992), 71–86.
  • W.M. Goldman: Complex Hyperbolic Geometry, Oxford Mathematical Monographs, Oxford University Press (1999).
  • W.M. Goldman and J. Parker: Dirichlet Polyhedra for Dihedral Groups Acting on Complex Hyperbolic Space, J. Geom. Anal. 91 (1999), 517–554.
  • K. Gongopadhyay and J.R. Parker: Reversible complex hyperbolic isometries, Linear Algebra Appl. 438 (2013), 2728–2739.
  • K. Gongopadhyay and C. Thomas: Decomposition of complex hyperbolic isometries by involutions, Linear Algebra Appl. 500 (2016), 63–76.
  • F. Knüppel and K. Nielsen: On products of two involutions in the orthogonal group of a vector space, Linear Algebra Appl. 94 (1987), 209–216.
  • C.P. Leo Jr: Real elements in small small cancellation groups, Math. Ann. 208 (1974), 279–293.
  • J.R. Parker:Notes on Complex Hyperbolic geometry, Preliminary version, 2003.
  • J. Paupert and P. Will: Real reflections, commutators and cross-ratios in complex hyperbolic space, Groups Geom. Dyn. 11 (2017), 311–352.
  • P. H. Tiep and A. E. Zalesski: Real conjugacy classes in algebraic groups and finite groups of Lie type, J. Group Theory 8 (2005), 291–315.
  • P. Will: The punctured torus and Lagrangian triangle groups in $\mathbf{PU}(2,1)$, J. Reine Angew. Math. 602 (2007), 95–121.
  • P. Will: Traces, cross-ratios and 2-generator subgroups of $\mathbf{SU}(2,1)$, Canad. J. Math. 61 (2009), 1407–1436.
  • M.J. Wonenburger: Transformations which are products of two involutions, J. Math. Mech. 16 (1966), 327–338.