Hyperbolic Actions and 2nd Bounded Cohomology of Subgroups of 𝖮𝗎𝗍(𝖥_{𝗇})

IF 2 4区数学 Q1 MATHEMATICS Memoirs of the American Mathematical Society Pub Date : 2023-12-01 DOI:10.1090/memo/1454

M. Handel, L. Mosher

{"title":"Hyperbolic Actions and 2nd Bounded Cohomology of Subgroups of 𝖮𝗎𝗍(𝖥_{𝗇})","authors":"M. Handel, L. Mosher","doi":"10.1090/memo/1454","DOIUrl":null,"url":null,"abstract":"<p>In this two part work we prove that for every finitely generated subgroup <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma greater-than sans-serif upper O sans-serif u sans-serif t left-parenthesis upper F Subscript n Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">Γ</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"sans-serif\">O</mml:mi>\n <mml:mi mathvariant=\"sans-serif\">u</mml:mi>\n <mml:mi mathvariant=\"sans-serif\">t</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>F</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\Gamma >{\\mathsf {Out}}(F_n)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, either <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\n <mml:semantics>\n <mml:mi mathvariant=\"normal\">Γ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is virtually abelian or <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript b Superscript 2 Baseline left-parenthesis normal upper Gamma semicolon double-struck upper R right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>H</mml:mi>\n <mml:mi>b</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"normal\">Γ</mml:mi>\n <mml:mo>;</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^2_b(\\Gamma ;{\\mathbb {R}})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> contains a vector space embedding of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l Superscript 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>ℓ</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\ell ^1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The method uses actions on hyperbolic spaces. In Part I we focus on the case of infinite lamination subgroups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\n <mml:semantics>\n <mml:mi mathvariant=\"normal\">Γ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>—those for which the set of all attracting laminations of all elements of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\n <mml:semantics>\n <mml:mi mathvariant=\"normal\">Γ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is an infinite set—using actions on free splitting complexes of free groups. In Part II we focus on finite lamination subgroups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\n <mml:semantics>\n <mml:mi mathvariant=\"normal\">Γ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and on the construction of useful new hyperbolic actions of those subgroups.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"114 12","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Memoirs of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1454","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 1

Abstract

In this two part work we prove that for every finitely generated subgroup Γ > O u t ( F n ) \Gamma >{\mathsf {Out}}(F_n) , either Γ \Gamma is virtually abelian or H b 2 ( Γ ; R ) H^2_b(\Gamma ;{\mathbb {R}}) contains a vector space embedding of ℓ 1 \ell ^1 . The method uses actions on hyperbolic spaces. In Part I we focus on the case of infinite lamination subgroups Γ \Gamma —those for which the set of all attracting laminations of all elements of Γ \Gamma is an infinite set—using actions on free splitting complexes of free groups. In Part II we focus on finite lamination subgroups Γ \Gamma and on the construction of useful new hyperbolic actions of those subgroups.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

𝖮𝗎𝗍(𝖥_{𝗇})子群的双曲作用和第二有界同调

在这两部分的工作中，我们证明了对于每一个有限生成的子群Γ > Out (F n) \Gamma >{\mathsf {Out}}(F_n)， Γ \Gamma要么是虚阿贝尔的，要么是h2 (Γ;R) H^2_b(\Gamma;{\mathbb {R}})包含一个向量空间嵌入，它包含一个向量空间嵌入。该方法在双曲空间上使用动作。在第一部分中，我们主要讨论无限层叠子群Γ \Gamma的情况——其中Γ \Gamma的所有元素的所有吸引层叠的集合是一个无限集——利用自由群的自由分裂配合物上的作用。在第二部分中，我们重点讨论了有限层合子群Γ \Gamma以及这些子群的有用的新双曲作用的构造。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Memoirs of the American Mathematical Society 数学-数学

CiteScore

3.50

自引率

5.30%

发文量

审稿时长

>12 weeks

期刊介绍： Memoirs of the American Mathematical Society is devoted to the publication of research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers or groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the AMS. To be accepted by the editorial board, manuscripts must be correct, new, and significant. Further, they must be well written and of interest to a substantial number of mathematicians.

期刊最新文献

Classification of 𝒪_{∞}-Stable 𝒞*-Algebras Angled Crested Like Water Waves with Surface Tension II: Zero Surface Tension Limit Hyperbolic Actions and 2nd Bounded Cohomology of Subgroups of 𝖮𝗎𝗍(𝖥_{𝗇}) Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩 The Generation Problem in Thompson Group 𝐹