2004 Fiscal Year Final Research Report Summary
Research on various geometric structures on 3-manifolds
| Project/Area Number |
15540073
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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 |
Geometry
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| Research Institution | Nara Women's University |
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Principal Investigator |
KOBAYASHI Tsuyoshi Nara Women's University, Science, Prof., 理学部, 教授 (00186751)
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| Co-Investigator(Kenkyū-buntansha) |
YAMASHITA Yasushi Nara Women's University, Science, Assist Prof., 理学部, 講師 (70239987)
KATAGIRI Minnyou Nara Women's University, Science, Associate Prof., 理学部, 助教授 (60263422)
ICHIHARA Kazuhiro Osaka Sangyo Univ., General Education, Assist Prof., 教養部, 講師 (00388357)
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| Project Period (FY) |
2003 – 2004
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| Keywords | 3-manifolds / Heegaard splittings / knots / tunnel number / Haken manifold / automatic group |
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| Research Abstract |
1.Morimoto's Conjecture on the tunnel numbers of composite knots in 3-manifolds
Let t(K) be the tunnel number of a knot K in a 3-manifold. Suppose for m-small knots K_1,…,K_n, the super additivity of tunnel number does not hold for #^n_<i=1> K_i, Then we proved that there exists a subset I of {1,【triple bond】,n} such that #_<i∈1>/K_i admits a primitive meridian.
2.The growth rate of tunnel number of knots
For a knot K in a 3-manifold M, we defined a numerical invariant called the growth rate of the tunnel numbers of K, and proved the following.
Suppose that the Heegaard genus of K is greater than the Heegaard genus of M. Then the growth rate of the tunnel numbers of K is less than 1.
3.Gersten's Problem for automatic group
Gersten posed the following problem "Each automatic group is eithr (1)a finite group, (2)contains free abelian group of rank 2,or (3)a word hyperbolic group." We showed that for a class of automatic group (called n-starred groups) this problem is solved affirmatively.
4.Heegaard gradients Seifert fibered spaces
We completely determined for which Seifert fibered space, the Heegaard gradient vanish.
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