{"title":"薄双曲反射群","authors":"Nikolay Bogachev, Alexander Kolpakov","doi":"10.1112/blms.70108","DOIUrl":null,"url":null,"abstract":"<p>We study a family of Zariski dense finitely generated discrete subgroups of <span></span><math>\n <semantics>\n <mrow>\n <mi>Isom</mi>\n <mo>(</mo>\n <msup>\n <mi>H</mi>\n <mi>d</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{Isom}(\\mathbb {H}^d)$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>⩾</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$d \\geqslant 2$</annotation>\n </semantics></math>, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application, we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a nonreflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in <span></span><math>\n <semantics>\n <mrow>\n <mi>Isom</mi>\n <mo>(</mo>\n <msup>\n <mi>H</mi>\n <mi>d</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{Isom}(\\mathbb {H}^d)$</annotation>\n </semantics></math>, for any <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>⩾</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$d \\geqslant 2$</annotation>\n </semantics></math>. Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2498-2508"},"PeriodicalIF":0.9000,"publicationDate":"2025-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70108","citationCount":"0","resultStr":"{\"title\":\"Thin hyperbolic reflection groups\",\"authors\":\"Nikolay Bogachev, Alexander Kolpakov\",\"doi\":\"10.1112/blms.70108\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study a family of Zariski dense finitely generated discrete subgroups of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Isom</mi>\\n <mo>(</mo>\\n <msup>\\n <mi>H</mi>\\n <mi>d</mi>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathrm{Isom}(\\\\mathbb {H}^d)$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>⩾</mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation>$d \\\\geqslant 2$</annotation>\\n </semantics></math>, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application, we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a nonreflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Isom</mi>\\n <mo>(</mo>\\n <msup>\\n <mi>H</mi>\\n <mi>d</mi>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathrm{Isom}(\\\\mathbb {H}^d)$</annotation>\\n </semantics></math>, for any <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>⩾</mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation>$d \\\\geqslant 2$</annotation>\\n </semantics></math>. Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"57 8\",\"pages\":\"2498-2508\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2025-06-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70108\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70108\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70108","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
We study a family of Zariski dense finitely generated discrete subgroups of , , defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application, we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a nonreflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in , for any . Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable.
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