Projective deformations of hyperbolic Coxeter 3-orbifolds

S Choi; CD Hodgson; GS Lee

Journal article

Projective deformations of hyperbolic Coxeter 3-orbifolds

S Choi, CD Hodgson, GS Lee

Geometriae Dedicata | SPRINGER | Published : 2012

DOI: 10.1007/s10711-011-9650-8

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Abstract

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahed..

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University of Melbourne Researchers

Craig Hodgson Author

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Awarded by Australian Research Council

Funding Acknowledgements

This work was carried out while two of the authors were visiting the Department of Mathematics and Statistics of the University of Melbourne; one of the authors also visited the Department of Mathematical and Computing Sciences in Tokyo Institute of Technology. We thank both institutions very much for their kind hospitality towards the authors. We have benefited much from discussions with Yves Benoist, Bill Goldman, Misha Kapovich, who studied this kind of question before, Sadayoshi Kojima, Daryl Cooper, and Steve Kerckhoff. We also very much appreciate the help and support of many mathematicians and others whose names are not mentioned here. We also thank the referee for interesting comments which led us to study projective bendings and to prove Theorem 10. S. Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2009-0057445). C. D. Hodgson was partially supported by Australian Research Council grants DP0663399 and DP1095760. G.-S. Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2008-621-C00003).