{"title":"尖角双曲表面上的电流和密度性质","authors":"Dounnu Sasaki","doi":"10.4171/GGD/688","DOIUrl":null,"url":null,"abstract":"The space $\\mathrm{GC} (\\Sigma)$ of geodesic currents on a hyperbolic surface $\\Sigma$ can be considered as a completion of the set of weighted closed geodesics on $\\Sigma$ when $\\Sigma$ is compact, since the set of rational geodesic currents on $\\Sigma$, which correspond to weighted closed geodesics, is a dense subset of $\\mathrm{GC}(\\Sigma )$. We prove that even when $\\Sigma$ is a cusped hyperbolic surface with finite area, $\\mathrm{GC}(\\Sigma )$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $\\Sigma$ but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to $\\mathrm{GC}(\\Sigma )$. To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Currents on cusped hyperbolic surfaces and denseness property\",\"authors\":\"Dounnu Sasaki\",\"doi\":\"10.4171/GGD/688\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The space $\\\\mathrm{GC} (\\\\Sigma)$ of geodesic currents on a hyperbolic surface $\\\\Sigma$ can be considered as a completion of the set of weighted closed geodesics on $\\\\Sigma$ when $\\\\Sigma$ is compact, since the set of rational geodesic currents on $\\\\Sigma$, which correspond to weighted closed geodesics, is a dense subset of $\\\\mathrm{GC}(\\\\Sigma )$. We prove that even when $\\\\Sigma$ is a cusped hyperbolic surface with finite area, $\\\\mathrm{GC}(\\\\Sigma )$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $\\\\Sigma$ but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to $\\\\mathrm{GC}(\\\\Sigma )$. To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.\",\"PeriodicalId\":55084,\"journal\":{\"name\":\"Groups Geometry and Dynamics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2020-11-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Groups Geometry and Dynamics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/GGD/688\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Groups Geometry and Dynamics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/GGD/688","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Currents on cusped hyperbolic surfaces and denseness property
The space $\mathrm{GC} (\Sigma)$ of geodesic currents on a hyperbolic surface $\Sigma$ can be considered as a completion of the set of weighted closed geodesics on $\Sigma$ when $\Sigma$ is compact, since the set of rational geodesic currents on $\Sigma$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(\Sigma )$. We prove that even when $\Sigma$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(\Sigma )$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $\Sigma$ but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to $\mathrm{GC}(\Sigma )$. To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.
期刊介绍:
Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.
Topics covered include:
geometric group theory;
asymptotic group theory;
combinatorial group theory;
probabilities on groups;
computational aspects and complexity;
harmonic and functional analysis on groups, free probability;
ergodic theory of group actions;
cohomology of groups and exotic cohomologies;
groups and low-dimensional topology;
group actions on trees, buildings, rooted trees.