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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Finiteness of polyhedral decompositions of cusped hyperbolic manifolds obtained by the Epstein-Penner's method


Author: Hirotaka Akiyoshi
Journal: Proc. Amer. Math. Soc. 129 (2001), 2431-2439
MSC (2000): Primary 51M10; Secondary 57M50
Published electronically: December 28, 2000
MathSciNet review: 1823928
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Abstract:

Epstein and Penner give a canonical method of decomposing a cusped hyperbolic manifold into ideal polyhedra. The decomposition depends on arbitrarily specified weights for the cusps. From the construction, it is rather obvious that there appear at most a finite number of decompositions if the given weights are slightly changed. However, since the space of weights is not compact, it is not clear whether the total number of such decompositions is finite. In this paper we prove that the number of polyhedral decompositions of a cusped hyperbolic manifold obtained by the Epstein-Penner's method is finite.


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Additional Information

Hirotaka Akiyoshi
Affiliation: Graduate School of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan
Email: akiyoshi@math.kyushu-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05829-9
PII: S 0002-9939(00)05829-9
Keywords: Hyperbolic manifold, canonical decomposition, ideal polyhedral decomposition
Received by editor(s): May 5, 1999
Received by editor(s) in revised form: December 21, 1999
Published electronically: December 28, 2000
Communicated by: Christopher Croke
Article copyright: © Copyright 2000 American Mathematical Society