A study of various complexities of pseudo-Anosov maps and hyperbolic fibered 3-manifolds

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03299
• 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: Osaka University
• Principal Investigator: Kin Eiko 大阪大学, 全学教育推進機構, 教授 (80378554)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 写像類群 / 擬アノソフ / 組ひも群 / エントロピー / 曲線グラフ / 漸近的移動距離

Outline of Final Research Achievements

We studied two invariants of pseudo-Anosov elements in the mapping class group. One is the entropy which is the translation length of the pseudo-Anosov element on the Teichmuller space. The other is the asymptotic translation length of the pseudo-Anosov element on the curve complex. (1) We gave a new construction of pseudo-Anosov braids with small normalized entropies. As an application, we determine asymptotic behaviors of minimal entropies of pseudo-Anosov elements in several subgroups (or several subsets) of mapping class groups. (Joint with Hirose, and Hirose Iguchi, Koda)(2) Given a fibered 3-manifold together with the fibered face, we give a general upper bound of asymptotic translation lengths of pseudo-Anosov monodromies associated with the fibered classes in the fibered cone.
(Joint with Baik, Shin and Wu)

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

曲面の写像類群の大部分は擬アノソフ写像類である. 擬アノソフ写像類の研究は力学系理論, ３次元多様体論, 双曲幾何学などのいくつかの分野と密接に関連する. 擬アノソフ写像類の代表的な不変量(エントロピー, 漸近的移動距離, 写像トーラスの体積)とこれらの不変量の関係の研究は位相幾何学(特に写像類群の研究)において基本的なテーマであり, それ故に学術的意義がある.