{"title":"Continuous deformation of the Bowen-Series map associated to a cocompact triangle group","authors":"","doi":"10.1007/s10711-024-00887-2","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on <span> <span>\\({\\mathbb {S}}^1\\)</span> </span> tightly associated with this group. In general, their fundamental domain enjoys what has since been called both the ‘extension property’ and the ‘even corners property’. We determine the exact set of signatures for cocompact triangle groups for which this property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"32 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-04-02","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/s10711-024-00887-2","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
In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on \({\mathbb {S}}^1\) tightly associated with this group. In general, their fundamental domain enjoys what has since been called both the ‘extension property’ and the ‘even corners property’. We determine the exact set of signatures for cocompact triangle groups for which this property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.
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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