{"title":"非强 Howson 群","authors":"Qiang Zhang, Dongxiao Zhao","doi":"arxiv-2409.09567","DOIUrl":null,"url":null,"abstract":"A group $G$ is called a Howson group if the intersection $H\\cap K$ of any two\nfinitely generated subgroups $H, K<G$ is again finitely generated, and called a\nstrongly Howson group when a uniform bound for the rank of $H\\cap K$ can be\nobtained from the ranks of $H$ and $K$. Clearly, every strongly Howson group is\na Howson group, but it is unclear in the literature whether the converse is\ntrue. In this note, we show that the converse is not true by constructing the\nfirst Howson groups which are not strongly Howson.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"207 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Howson groups which are not strongly Howson\",\"authors\":\"Qiang Zhang, Dongxiao Zhao\",\"doi\":\"arxiv-2409.09567\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A group $G$ is called a Howson group if the intersection $H\\\\cap K$ of any two\\nfinitely generated subgroups $H, K<G$ is again finitely generated, and called a\\nstrongly Howson group when a uniform bound for the rank of $H\\\\cap K$ can be\\nobtained from the ranks of $H$ and $K$. Clearly, every strongly Howson group is\\na Howson group, but it is unclear in the literature whether the converse is\\ntrue. In this note, we show that the converse is not true by constructing the\\nfirst Howson groups which are not strongly Howson.\",\"PeriodicalId\":501037,\"journal\":{\"name\":\"arXiv - MATH - Group Theory\",\"volume\":\"207 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Group Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09567\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Group Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09567","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}