Geometry and topology of 3-manifolds II

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The purpose of this research project is to see the interaction of various geometric structures on 3-manifolds, to find principal idea behind them such as laws in physics and to promote the study of 3-manifolds interacting geometry and topology interdisciplinary. The current project based on the primary one that terminated on March 2003 was set up for 4 years and we have spent 3 years so far. Then we have decided to pre-renew the project by shifting the stress more on invariants because it seems to be necessary to do so in the light of recent progress on the current working subject. Fortunately, the new project was awarded as a continuing one and thus we here report our activity of the last 3 years.In the last 3 years, we, the member of research group and collaborators, have promoted the project by doing research communication, organizing meetings and publishing research reports. As conclusion, we, the member of research group, have obtained a plenty of significant progress and simultaneously contributed the progress of this field. The research results were presented in the academic meetings including international ones. The details including one's by collaborators can be found in the attached report. In summary, according to the purpose of the project, we have darified what is needed for geometric methods to study topology of 3-manioflds more and more.Looking at last 3 years, we must notice historical progress such as the solution of the ending lamination conjecture by Minsky-Brock-Canary the solution of Marden's conjecture by Agol and Calegari-Gabai and the approach to the geometrization by Hamilton and Perelman, which at least includes the solution of Poincare conjecture. Based on these, we set up the post geometrization by renewing our project with entitling "The geometry and invariants of 3-manifolds".