{"title":"Cayley extensions of maniplexes and polytopes","authors":"Gabe Cunningham , Elías Mochán , Antonio Montero","doi":"10.1016/j.jcta.2024.106000","DOIUrl":null,"url":null,"abstract":"<div><div>A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define <span><math><mi>M</mi></math></span> to be a <em>Cayley extension</em> of <span><math><mi>K</mi></math></span> if the facets of <span><math><mi>M</mi></math></span> are isomorphic to <span><math><mi>K</mi></math></span> and if some subgroup of the automorphism group of <span><math><mi>M</mi></math></span> acts regularly on the facets of <span><math><mi>M</mi></math></span>. We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106000"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524001390","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define to be a Cayley extension of if the facets of are isomorphic to and if some subgroup of the automorphism group of acts regularly on the facets of . We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.