Translations and Associated Dirichlet Polyhedra in Complex Hyperbolic Space |
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Authors: | Po-Hsun Hsieh |
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Institution: | (1) Department of Mathematics, National Chung Cheng University, Minhsiung, Chiayi, 621, Taiwan |
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Abstract: | We study a nonidentity transvection (i.e. (strictly) hyperbolic isometry) or nonidentity Heisenberg translation f of complex hyperbolic space H
n and a Dirichlet polyhedron P of the cyclic group f. We have four main results: (a) If z & in H
n and the axis of a nonidentity transvection are not complex collinear, then, roughly speaking, any two distinct 'naturally arising' geodesics passing through z are not complex collinear. (b) If g is also a transvection or Heisenberg translation of H
n and z & in H
n such that f(z)=g(z) and f
–1(z)=g
–1(z), then f=g. (c) We classify all this kind of polyhedra up to congruence in H
n. (d) We obtain an equivalent condition for P to be cospinal (which means that the complex spines of the two sides of P coincide) in terms of the distance of the spines of the two sides of P. |
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Keywords: | Dirichlet polyhedra Heisenberg translation hyperbolic space |
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