2000 Fiscal Year Final Research Report Summary
STUDY ON DISCRETE SUBGROUPS OF PU(1, n ; C)
| Project/Area Number |
11640192
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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) |
SHIMENO Nobukazu OKAYAMA UNIVERSITY OF SCIENCE, ASSISTANT PROFESSOR, 理学部, 助教授 (60254140)
TAKENAKA Shigeo OKAYAMA UNIVERSITY OF SCIENCE, PROFESSOR, 理学部, 教授 (80022680)
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| Project Period (FY) |
1999 – 2000
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| Keywords | COMPLEX HYPERBOLIC SPACE / PU(1, n ; C) / DISCRETE GROUP / ISOMETRIC SPHERE / FORD REGION / DIRICHLET POLYHEDRON |
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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. Given a discrete subgroup of Mobius transformations containing a parabolic element with fixed point ∞, a classical result, which is called Shimizu's lemma, gives a uniform bound on the radii of isometric circles of those elements of the group not fixing ∞. Recently Parker has shown that if a discrete subgroup G of PU(1, n ; C) contains a Heisenberg translation g, then any element of G not sharing a fixed point with g has an isometric sphere whose radius is bounded above by a function of the translation length of g at its centers. This Parker's theorem is considered as a generalization of Shimizu's lemma. Basmajian and Miner have independently obtained qualitatively similar results for discrete subgroups of PU(1,2 ; C) by using their stable basin theorem. First we improve the stable basin theorem, and second we show that under some conditions Parker's theorem yields the discreteness condition of Basmajian and Miner for groups with a Heisenberg translation. The latter answers a question posed in Parker's paper.
2) Let G be a discrete subgroup of PU(1, n ; C). For a boundary point y of the Siegel domain, we define the generalized isometric sphere I^y (f) of an element f of PU(1, n ; C). By using the generalized isometric spheres of elements of G, we construct a fundamental domain P^y (G) for G, which is regarded as a generalization of the Ford domain. And we show that the Dirichlet polyhedron D(w) for G with center w convereges to P^y(G) as w → y.
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