Dirichlet polyhedra for cyclic groups in complex hyperbolic space
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- by Mark B. Phillips PDF
- Proc. Amer. Math. Soc. 115 (1992), 221-228 Request permission
Abstract:
We prove that the Dirichlet fundamental polyhedron for a cyclic group generated by a unipotent or hyperbolic element $\gamma$ acting on complex hyperbolic $n$-space centered at an arbitrary point $w$ is bounded by the two hypersurfaces equidistant from the pairs $w,\gamma w$ and $w,{\gamma ^{ - 1}}w$ respectively. The proof relies on a convexity property of the distance to an isometric flow containing $\gamma$.References
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
- B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245–317. MR 1218098, DOI 10.1006/jfan.1993.1052
- S. S. Chen and L. Greenberg, Hyperbolic spaces, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 49–87. MR 0377765
- Paul E. Ehrlich and Hans-Christoph Im Hof, Dirichlet regions in manifolds without conjugate points, Comment. Math. Helv. 54 (1979), no. 4, 642–658. MR 552682, DOI 10.1007/BF02566298
- D. B. A. Epstein, Complex hyperbolic geometry, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 93–111. MR 903851 W. M. Goldman, Introduction to complex hyperbolic geometry, lecture notes for course at the Univ. of Maryland, January 1990.
- Troels Jørgensen, On cyclic groups of Möbius transformations, Math. Scand. 33 (1973), 250–260 (1974). MR 348103, DOI 10.7146/math.scand.a-11487
- J. J. Stoker, Differential geometry, Pure and Applied Mathematics, Vol. XX, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969. MR 0240727
- G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876 M. B. Phillips, Dirichlet polyhedra in complex hyperbolic space, Doctoral Dissertation, Univ. of Maryland at College Park, 1990.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 221-228
- MSC: Primary 32H20; Secondary 51M10, 51M20
- DOI: https://doi.org/10.1090/S0002-9939-1992-1107276-1
- MathSciNet review: 1107276