{"title":"三角形群的可溶性商","authors":"Marston D.E. Conder, Darius W. Young","doi":"10.1016/j.jalgebra.2024.10.012","DOIUrl":null,"url":null,"abstract":"<div><div>This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for ‘small’ genus), by showing that every non-perfect hyperbolic ordinary triangle group <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>=</mo><mo>〈</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>y</mi><mspace></mspace><mo>|</mo><mspace></mspace><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mrow><mi>r</mi></mrow></msup><mo>=</mo><mn>1</mn><mspace></mspace><mo>〉</mo></math></span> has a smooth finite soluble quotient of derived length <em>c</em> for some <span><math><mi>c</mi><mo>≤</mo><mn>3</mn></math></span>, and infinitely many such quotients of derived length <em>d</em> for every <span><math><mi>d</mi><mo>></mo><mi>c</mi></math></span>.</div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Soluble quotients of triangle groups\",\"authors\":\"Marston D.E. Conder, Darius W. Young\",\"doi\":\"10.1016/j.jalgebra.2024.10.012\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for ‘small’ genus), by showing that every non-perfect hyperbolic ordinary triangle group <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>=</mo><mo>〈</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>y</mi><mspace></mspace><mo>|</mo><mspace></mspace><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mrow><mi>r</mi></mrow></msup><mo>=</mo><mn>1</mn><mspace></mspace><mo>〉</mo></math></span> has a smooth finite soluble quotient of derived length <em>c</em> for some <span><math><mi>c</mi><mo>≤</mo><mn>3</mn></math></span>, and infinitely many such quotients of derived length <em>d</em> for every <span><math><mi>d</mi><mo>></mo><mi>c</mi></math></span>.</div></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021869324005568\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869324005568","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for ‘small’ genus), by showing that every non-perfect hyperbolic ordinary triangle group has a smooth finite soluble quotient of derived length c for some , and infinitely many such quotients of derived length d for every .
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