{"title":"Prym representations of the handlebody group","authors":"","doi":"10.1007/s10711-024-00911-5","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Let <em>S</em> be an oriented, closed surface of genus <em>g</em>. The mapping class group of <em>S</em> is the group of orientation preserving homeomorphisms of <em>S</em> modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let <em>V</em> be a genus <em>g</em> handlebody with boundary <em>S</em>. The handlebody group is the subgroup of those mapping classes of <em>S</em> that extend over <em>V</em>. The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"47 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometriae Dedicata","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10711-024-00911-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let S be an oriented, closed surface of genus g. The mapping class group of S is the group of orientation preserving homeomorphisms of S modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let V be a genus g handlebody with boundary S. The handlebody group is the subgroup of those mapping classes of S that extend over V. The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.
S 的映射类群是 S 的方向保持同构群。1997 年，Looijenga 引入了 Prym 表示，它是映射类群的虚拟表示，取决于一个有限的无性群。让 V 是具有边界 S 的 g 属手柄体。手柄体群是 S 的映射类在 V 上延伸的子群。在此，我们将 Prym 表示限定于柄体群，并进一步限定于扭转群。我们将确定循环情况下的表示的图像。
期刊介绍：
Geometriae Dedicata concentrates on geometry and its relationship to topology, group theory and the theory of dynamical systems.
Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and related areas. Features of the journal will include:
A fast turn-around time for articles.
Special issues centered on specific topics.
All submitted papers should include some explanation of the context of the main results.
Geometriae Dedicata was founded in 1972 on the initiative of Hans Freudenthal in Utrecht, the Netherlands, who viewed geometry as a method rather than as a field. The present Board of Editors tries to continue in this spirit. The steady growth of the journal since its foundation is witness to the validity of the founder''s vision and to the success of the Editors'' mission.
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