A study of Entropies of pseudo-Anosov mapping classes and hyperbolic fibered 3-manifolds

2015 Fiscal Year Final Research Report

• Project/Area Number: 24740039
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: Kin Eiko 大阪大学, 理学(系)研究科(研究院), 准教授 (80378554)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 位相幾何 / 双曲多様体 / ファイバー多様体 / 写像類群 / 擬アノソフ / エントロピー / 群の不変順序

Outline of Final Research Achievements

We consider the mapping class group of an orientable surface. Given a mapping class, we can build its mapping torus which is a fibered 3-manifold. It is known by Thurston that a mapping class is　pseudo-Anosov (pA) if and only if its mapping torus is hyperbolic. If we fix a fibered 3-manifold M with the first Betti number greater than 1, there are infinitely many ways to represent M as mapping tori. I consider the so-called magic 3-manifold N which is a hyperbolic 3-manifold with 3 cusps.
I explicitly construct every pA mapping class whose mapping torus is homeomorphic N. By using these pA mapping classes, I study some properties on bi-orderings of the fundamental groups of some hyperbolic fibered 3-manifolds.

Free Research Field

位相幾何学