{"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}
引用次数: 0
Abstract
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.