Development of software to and researches in hyperbolic geometry and kleinnian group

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In this project, we have developed a research tool OPTi for researchers in hyperbolic geometry and Kleinnian groups, and improved it in various ways. The results obtained in the project have always been open to the public through the OPTi Home Page (RUL: http://vivaldi.ics.nara-wu.ac.jp/〜wada/OPTi/index.html). We gave talks related to OPTi in many conferences and workshops including The 5th International Conference in Clifford Algebra, Mexico, 1999, Workshop on Computations in Group Theory and Geometry, University of Warwick, 1999, Workshop on Kleinian Groups and Hyperbolic 3-Manifolds, University of Warwick, 2001. We also organized workshops "Punctured tori and 2-bridge knots ", Nara Women's University, 2000 and "Studies on Geometric 3-orbifolds", Nara Women's Univeristy, 2000, and heard comments and 1 requests about OPTi from many researchers. As a result, OPTi has become much more useful and is much easier to use than the first version.Meanwhile, in 2001, in cooperation with Yamashita, Sugawa and Komori, we applied the algorithm for drawing the Ford regions used in OPTi to the Bers slice, and obtained pictures of the exotic components for the first time in the world. The algorithm used therein was later implemented into OPTi Version 3.30 to give real-time drawing of the deformations of various kinds of slice pictures.