{"title":"关于k-大地测量图和群","authors":"M. Elder, Adam Piggott, K. Townsend","doi":"10.1142/s0218196723500534","DOIUrl":null,"url":null,"abstract":"We call a graph $k$-geodetic, for some $k\\geq 1$, if it is connected and between any two vertices there are at most $k$ geodesics. It is shown that any hyperbolic group with a $k$-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centraliser of any infinite order element is an infinite cyclic group. These results were known previously only in the case that $k=1$. A key tool used to develop the theorem is a new graph theoretic result concerning ``ladder-like structures'' in a $k$-geodetic graph.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On k-geodetic graphs and groups\",\"authors\":\"M. Elder, Adam Piggott, K. Townsend\",\"doi\":\"10.1142/s0218196723500534\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We call a graph $k$-geodetic, for some $k\\\\geq 1$, if it is connected and between any two vertices there are at most $k$ geodesics. It is shown that any hyperbolic group with a $k$-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centraliser of any infinite order element is an infinite cyclic group. These results were known previously only in the case that $k=1$. A key tool used to develop the theorem is a new graph theoretic result concerning ``ladder-like structures'' in a $k$-geodetic graph.\",\"PeriodicalId\":13756,\"journal\":{\"name\":\"International Journal of Algebra and Computation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-11-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Algebra and Computation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0218196723500534\",\"RegionNum\":2,\"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":"International Journal of Algebra and Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0218196723500534","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
We call a graph $k$-geodetic, for some $k\geq 1$, if it is connected and between any two vertices there are at most $k$ geodesics. It is shown that any hyperbolic group with a $k$-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centraliser of any infinite order element is an infinite cyclic group. These results were known previously only in the case that $k=1$. A key tool used to develop the theorem is a new graph theoretic result concerning ``ladder-like structures'' in a $k$-geodetic graph.
期刊介绍:
The International Journal of Algebra and Computation publishes high quality original research papers in combinatorial, algorithmic and computational aspects of algebra (including combinatorial and geometric group theory and semigroup theory, algorithmic aspects of universal algebra, computational and algorithmic commutative algebra, probabilistic models related to algebraic structures, random algebraic structures), and gives a preference to papers in the areas of mathematics represented by the editorial board.