{"title":"双曲群的子群，有限性和复双曲格","authors":"Claudio Llosa Isenrich, Pierre Py","doi":"10.1007/s00222-023-01223-3","DOIUrl":null,"url":null,"abstract":"Abstract We prove that in a cocompact complex hyperbolic arithmetic lattice $\\Gamma < {\\mathrm{PU}}(m,1)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Γ</mml:mi> <mml:mo><</mml:mo> <mml:mi>PU</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:math> of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type $\\mathscr{F}_{m-1}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> but not of type $\\mathscr{F}_{m}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> . This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices\",\"authors\":\"Claudio Llosa Isenrich, Pierre Py\",\"doi\":\"10.1007/s00222-023-01223-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We prove that in a cocompact complex hyperbolic arithmetic lattice $\\\\Gamma < {\\\\mathrm{PU}}(m,1)$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>Γ</mml:mi> <mml:mo><</mml:mo> <mml:mi>PU</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:math> of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type $\\\\mathscr{F}_{m-1}$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> but not of type $\\\\mathscr{F}_{m}$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> . This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2023-10-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00222-023-01223-3\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00222-023-01223-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 7
摘要
摘要证明了紧复双曲算术格$\Gamma <{\ mathm {PU}}(m,1)$ Γ <最简单类型的PU (m, 1)，足够深的有限索引子群承认具有核类型为$\mathscr{F}_{m-1}$ F m-1但不具有核类型为$\mathscr{F}_{m}$ F m的大量同态。这提供了许多有限表示的双曲群的非双曲子群，并回答了Brady的一个老问题。我们的方法也证明了非球面Kähler流形的辛格猜想的一个特例。
Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices
Abstract We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma < {\mathrm{PU}}(m,1)$ Γ<PU(m,1) of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type $\mathscr{F}_{m-1}$ Fm−1 but not of type $\mathscr{F}_{m}$ Fm . This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.
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