The geometry of character variety and classification of arithmetic Kleinian groups

2022 Fiscal Year Final Research Report

• Project/Area Number: 20K03612
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Nara Women's University
• Principal Investigator: Yamashita Yasushi 奈良女子大学, 自然科学系, 教授 (70239987)
• Project Period (FY): 2020-04-01 – 2023-03-31
• Keywords: 双曲幾何学 / クライン群

Outline of Final Research Achievements

Hyperbolic geometry is an important geometric structure for manifolds of dimension 2 and 3. To understand this structure, we studied index manifolds of the fundamental group of 2-dimensional manifolds. In particular, we studied algebraically and geometrically interesting Klein groups generated by two elements of SL(2,C), which are called arithmetic Klein groups when the number of degrees of the generators is finite.
As an approach to the research, we adopted a method that uses computer experiments in particular, and succeeded in effectively dealing with a problem that was difficult to deal with by conventional methods.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

現代の位相幾何学における主要な研究対象である図形に多様体とよばれるものがあり、それらがどのような形の変形を許容するのかという問題にアプローチすることは、数学の研究を進める上で基本的な意義がある。さらに、算術性などの代数的な手法や写像類群の作用という力学系との関係を明らかにすることにより、分野間の新たなつながりの解明に貢献した。また、本研究は手法としては計算機実験を特徴としており、計算機の応用領域を数理科学に拡げるという形での意義もある。