Study of rigidity of foliations based on global geometry of leaves

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03620
• 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: Ritsumeikan University
• Principal Investigator: Nozawa Hiraku 立命館大学, 理工学部, 准教授 (80706557)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 葉層構造 / 力学系 / カオス / タイル張り / トポロジー / 群作用 / 情報幾何 / 剛性

Outline of Final Research Achievements

I conducted research on the rigidity of foliations from various viewpoints. We studied the surface group actions on the circle to obtain rigidity results of the harmonic measures of their suspension foliations using Thurston's connection. As an application, we provided alternative proofs for well-known rigidity theorems, such as the Milnor-Wood inequality and the Matsumoto rigidity theorem generalized by Burger-Iozzi-Wienhard to surfaces with cusps. Additionally, I constructed new examples of Delone sets (uniformly scattered point sets) in symmetric spaces of non-compact type by generalizing the cut-and-project method. Furthermore, we constructed new examples of equicontinuous group actions on the Cantor set and investigated the chaotic properties of colored graphs from a topological viewpoint.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

葉層構造は群作用，偏微分方程式などの自然現象をとらえるための理論において現れる幾何的対象である．本研究では，葉層構造の中でも特に対称性の高いものやカオス的な性質を持つものに注目し，様々な研究を行った．これらの葉層構造はリーマン面の幾何学などの古典的数学にも現れ，幾何学，代数学，複素解析や物理学などにおいて研究されている．本研究の結果はこれらの関連分野の研究を深めるために意義があると考えられる．