2002 Fiscal Year Final Research Report Summary
Study on Fundamental Regions for Discrete Subgroups of PU(1,n;C)
| Project/Area Number |
13640198
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Okayama University of Science |
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Principal Investigator |
KAMIYA Shigeyasu Okayama University of Science, Professor, 工学部, 教授 (80122381)
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| Co-Investigator(Kenkyū-buntansha) |
MURAKAMI Satoru Okayama University of Science, Professor, 理学部, 教授 (40123963)
TAKENAKA Shigeo Okayama University of Science, Professor, 理学部, 教授 (80022680)
SHIMENO Nobukazu Okayama University of Science, Assistant Professor, 理学部, 助教授 (60254140)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Unitary group / Discrete group / Fundamental region / Heisenberg translation |
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| Research Abstract |
1) In the study of discrete groups, it is important to find conditions for a group to be discrete. In the case of Mobius transformations Jorgensen's inequality is very famous. Kamiya, Basmajian-Miner and Parker have discussed the discreteness conditions for subgroups of PU(1,2;C) with parabolic elements. We give the relationship between them and also generalize Basmajian-Miner's stable basin theorem to wider class of subgroups of PU(1,2;C). By using the cross-ratio in the boundary of complex hyperbolic space, we obtain analogues of Jorgensen's inequality for non-elementary groups generated by two elements, one of which is either loxodromic or boundary elliptic.
2) We define the generalized isometric of an element of PU(1,n;C). By using the generalized isometric spheres of elements of a discrete subgroup of PU(1,n;C), we construct a fundamental domain, which is regarded as a generalization of the Ford domain. And we show the connection between this generalized Ford domain and Dirichlet polyhedron. Now we are trying to use these fundamental regions for studying the deformation space.
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