Currents on cusped hyperbolic surfaces

2021 Fiscal Year Final Research Report

• Project/Area Number: 19K14539
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Gakushuin University (2021); Waseda University (2019-2020)
• Principal Investigator: Sasaki Dounnu 学習院大学, 理学部, 日本学術振興会特別研究員 (60822484)
• Project Period (FY): 2019-04-01 – 2022-03-31
• Keywords: 双曲曲面 / 測地カレント / サブセットカレント / 稠密性定理 / 交点数 / 曲面群 / 自由群

Outline of Final Research Achievements

The space GC(S) of geodesic currents on a hyperbolic surface S can be considered as a completion of the set of weighted closed geodesics on S when S is compact, since the set of rational geodesic currents on S, which correspond to weighted closed geodesics, is a dense subset of GC(S). I proved that even when S is a cusped hyperbolic surface with finite area, GC(S) has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on S but also to weighted geodesics connecting two cusps. In addition, I proved that the space of subset currents on a cusped hyperbolic surface, which is a generalization of geodesic currents, also has the denseness property of rational subset currents.

Free Research Field

双曲幾何学・幾何学的群論

Academic Significance and Societal Importance of the Research Achievements

双曲曲面上の測地カレントはThurstonによって導入された測度付き測地線層（単純閉測地線の完備化）の一般化としてBonahonによって導入された．双曲曲面の変形空間であるタイヒミュラー空間や写像類群の研究とも密接に関係する．近年ではMirzakhaniによって閉測地線の数え上げ問題への応用が見出された．ただし，測地カレントを閉測地線の完備化と見るためには稠密性定理を示す必要があるのだが，これまではコンパクトな双曲曲面の場合にしか示されていなかった．本研究によりカスプ付きの場合でも稠密性定理が示され，測地カレントの研究の基礎的な部分が整備されたと言える．