{"title":"几乎自由群组内定点的最终定点","authors":"André Carvalho","doi":"10.1093/qmath/haae032","DOIUrl":null,"url":null,"abstract":"We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $\\text{EvFix}(\\varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $\\text{End}(G)$, if φ is aperiodic, if $\\text{EvFix}(\\varphi)$ is finitely generated and if $\\text{EvFix}(\\varphi)$ is a normal subgroup. In the cases where $\\text{EvFix}(\\varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Eventually fixed points of endomorphisms of virtually free groups\",\"authors\":\"André Carvalho\",\"doi\":\"10.1093/qmath/haae032\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $\\\\text{EvFix}(\\\\varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $\\\\text{End}(G)$, if φ is aperiodic, if $\\\\text{EvFix}(\\\\varphi)$ is finitely generated and if $\\\\text{EvFix}(\\\\varphi)$ is a normal subgroup. In the cases where $\\\\text{EvFix}(\\\\varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.\",\"PeriodicalId\":54522,\"journal\":{\"name\":\"Quarterly Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-06-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quarterly Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1093/qmath/haae032\",\"RegionNum\":4,\"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":"Quarterly Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/qmath/haae032","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Eventually fixed points of endomorphisms of virtually free groups
We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $\text{EvFix}(\varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $\text{End}(G)$, if φ is aperiodic, if $\text{EvFix}(\varphi)$ is finitely generated and if $\text{EvFix}(\varphi)$ is a normal subgroup. In the cases where $\text{EvFix}(\varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.
期刊介绍:
The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.