2007 Fiscal Year Final Research Report Summary
The complex hyperbolic structures on the configuration spaces of points on the sphere and surface subgroup of mapping class groups
| Project/Area Number |
18540085
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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 |
YAMASHITA Yasushi Nara Women's University, Faculty of Science, Associate Professor (70239987)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Configuration space / Hyperbolic geometry / Kleinian groups |
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| Research Abstract |
(1) Structures of non-hyperbolic automatic groups
(Joint work with Y. Nakagawa, M. Tamura)
Let G be a finitely presented group. If G contains a Z + Z subgroup, then it is well known that G cannot be word hyperbolic. A natural question is that "is Z + Z the only obstruction for a finitely presented group to be word hyperbolic?" In other words, "if G does not contain any Z + Z subgroups, is it word hyperbolic?" Baumslag-Solitar groups are counter examples to this question. Thus it would be better to restrict our attention to some good class of groups. Here we focus on automatic groups. Note that Baumslag-Solitar groups are not automatic. Our problem is indicated in the list of open problems and attributed to Gersten. We call this problem "Gersten's problem".
Recall that the class of all automatic groups contains the class of all hyperbolic groups, all virtually abelian groups and all geometrically finite hyperbolic groups. A geometrically finite hyperbolic group is, in some sense, similar to hyperbolic groups, but it might contain a Z + Z subgroup. Thus the class of automatic groups is a nice target to consider the question mentioned before.
We define the notion of "n-tracks of length n", which suggests a clue of the existence of Z + Z subgroup and shows its existence in every non-hyperbolic automatic groups with mild conditions.
(2) The character variety of one-holed torus
(Joint work with S.P. Tan)
The quasifuchsian space of punctured torus groups is deeply studied by many people and some of the major conjectures on them are solved in the last decade. But, for general "one-holed" cases, not much is known. In this study, we produced computer software to investigate the character variety of one holed torus and were able to find many interasting phenomena
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