{"title":"穆勒-舒普定理的一般化和树状逆图","authors":"Emanuele Rodaro","doi":"10.1112/jlms.12903","DOIUrl":null,"url":null,"abstract":"<p>We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph <span></span><math>\n <semantics>\n <mi>Γ</mi>\n <annotation>$\\Gamma$</annotation>\n </semantics></math>, being quasi-isometric to a tree, or context-free in the sense of Muller–Schupp (finitely many end-cone up to end-isomorphism), or having the automorphism group <span></span><math>\n <semantics>\n <mrow>\n <mo>Aut</mo>\n <mo>(</mo>\n <mi>Γ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{Aut}(\\Gamma)$</annotation>\n </semantics></math> that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky–Schützenberger that is fundamental in solving a weaker version of a conjecture of Brough which also extends Muller and Schupp's result to the class of groups that are virtually finitely generated subgroups of the direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generalizations of the Muller–Schupp theorem and tree-like inverse graphs\",\"authors\":\"Emanuele Rodaro\",\"doi\":\"10.1112/jlms.12903\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph <span></span><math>\\n <semantics>\\n <mi>Γ</mi>\\n <annotation>$\\\\Gamma$</annotation>\\n </semantics></math>, being quasi-isometric to a tree, or context-free in the sense of Muller–Schupp (finitely many end-cone up to end-isomorphism), or having the automorphism group <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>Aut</mo>\\n <mo>(</mo>\\n <mi>Γ</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{Aut}(\\\\Gamma)$</annotation>\\n </semantics></math> that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky–Schützenberger that is fundamental in solving a weaker version of a conjecture of Brough which also extends Muller and Schupp's result to the class of groups that are virtually finitely generated subgroups of the direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12903\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12903","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们从两个方面扩展了穆勒和舒普对无上下文群的描述。我们首先证明,对于准传递逆图 Γ $\Gamma$ 而言,准等距于树、或穆勒-舒普意义上的无上下文(有限多个端锥到端异构)、或具有实际上自由的自动形态群 Aut ( Γ ) $\operatorname{Aut}(\Gamma)$ 都是等价条件。此外,我们还在前面的等价条件中加入了一个与乔姆斯基-舒岑伯格的表示定理类似的群论,它是解决布劳夫猜想的弱化版本的基础,而布劳夫猜想也将穆勒和舒普的结果扩展到了自由群的直积的虚拟有限生成子群这一类群。我们证明,这类群正是那些其文字问题是由准传递、树状逆图所接受的有限数量语言的交集的群。
Generalizations of the Muller–Schupp theorem and tree-like inverse graphs
We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph , being quasi-isometric to a tree, or context-free in the sense of Muller–Schupp (finitely many end-cone up to end-isomorphism), or having the automorphism group that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky–Schützenberger that is fundamental in solving a weaker version of a conjecture of Brough which also extends Muller and Schupp's result to the class of groups that are virtually finitely generated subgroups of the direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.