{"title":"直积的二元子群","authors":"M. Bridson","doi":"10.4171/lem/1057","DOIUrl":null,"url":null,"abstract":"We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\\em binary subgroups}, $B(\\Sigma,\\mu)<G_1\\times\\dots\\times G_m$. These full subdirect products require strikingly few generators. If each $G_i$ is finitely presented, $B(\\Sigma,\\mu)$ is finitely presented. When the $G_i$ are non-abelian limit groups (e.g. free or surface groups), the $B(\\Sigma,\\mu)$ provide new examples of finitely presented, residually-free groups that do not have finite classifying spaces and are not of Stallings-Bieri type. These examples settle a question of Minasyan relating different notions of rank for residually-free groups. Using binary subgroups, we prove that if $G_1,\\dots,G_m$ are perfect groups, each requiring at most $r$ generators, then $G_1\\times\\dots\\times G_m$ requires at most $r \\lfloor \\log_2 m+1 \\rfloor$ generators.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"122 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Binary subgroups of direct products\",\"authors\":\"M. Bridson\",\"doi\":\"10.4171/lem/1057\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\\\\em binary subgroups}, $B(\\\\Sigma,\\\\mu)<G_1\\\\times\\\\dots\\\\times G_m$. These full subdirect products require strikingly few generators. If each $G_i$ is finitely presented, $B(\\\\Sigma,\\\\mu)$ is finitely presented. When the $G_i$ are non-abelian limit groups (e.g. free or surface groups), the $B(\\\\Sigma,\\\\mu)$ provide new examples of finitely presented, residually-free groups that do not have finite classifying spaces and are not of Stallings-Bieri type. These examples settle a question of Minasyan relating different notions of rank for residually-free groups. Using binary subgroups, we prove that if $G_1,\\\\dots,G_m$ are perfect groups, each requiring at most $r$ generators, then $G_1\\\\times\\\\dots\\\\times G_m$ requires at most $r \\\\lfloor \\\\log_2 m+1 \\\\rfloor$ generators.\",\"PeriodicalId\":344085,\"journal\":{\"name\":\"L’Enseignement Mathématique\",\"volume\":\"122 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-02-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"L’Enseignement Mathématique\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4171/lem/1057\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"L’Enseignement Mathématique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/lem/1057","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\em binary subgroups}, $B(\Sigma,\mu)
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