Normal subgroups of big mapping class groups

Transactions of the American Mathematical Society, Series B Pub Date : 2021-10-15 DOI:10.1090/btran/108

Danny Calegari, Lvzhou Chen

{"title":"Normal subgroups of big mapping class groups","authors":"Danny Calegari, Lvzhou Chen","doi":"10.1090/btran/108","DOIUrl":null,"url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\">\n <mml:semantics>\n <mml:mi>S</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">S</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a surface and let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M o d left-parenthesis upper S comma upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Mod</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\operatorname {Mod}(S,K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be the mapping class group of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\">\n <mml:semantics>\n <mml:mi>S</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">S</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> permuting a Cantor subset <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K subset-of upper S\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>K</mml:mi>\n <mml:mo>⊂</mml:mo>\n <mml:mi>S</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">K \\subset S</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We prove two structure theorems for normal subgroups of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M o d left-parenthesis upper S comma upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Mod</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\operatorname {Mod}(S,K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.\n\n(Purity:) if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\">\n <mml:semantics>\n <mml:mi>S</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">S</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> has finite type, every normal subgroup of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M o d left-parenthesis upper S comma upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Mod</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\operatorname {Mod}(S,K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> either contains the kernel of the forgetful map to the mapping class group of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\">\n <mml:semantics>\n <mml:mi>S</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">S</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, or it is ‘pure’ — i.e. it fixes the Cantor set pointwise.\n\n(Inertia:) for any <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> element subset <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q\">\n <mml:semantics>\n <mml:mi>Q</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">Q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of the Cantor set, there is a forgetful map from the pure subgroup <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P upper M o d left-parenthesis upper S comma upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>PMod</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\operatorname {PMod}(S,K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M o d left-parenthesis upper S comma upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Mod</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\operatorname {Mod}(S,K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to the mapping class group of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper S comma upper Q right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>Q</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(S,Q)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> fixing <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q\">\n <mml:semantics>\n <mml:mi>Q</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">Q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> pointwise. If <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\n <mml:semantics>\n <mml:mi>N</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a normal subgroup of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M o d left-parenthesis upper S comma upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Mod</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\operatorname {Mod}(S,K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> contained in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P upper M o d left-parenthesis upper S comma upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>PMod</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\operatorname {PMod}(S,K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, its image <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper Q\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>N</mml:mi>\n <mml:mi>Q</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">N_Q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is likewise normal. We characterize exactly which finite-type normal subgroups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper Q\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>N</mml:mi>\n <mml:mi>Q</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">N_Q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> arise this way.\n\nSeveral applications and numerous examples are also given.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"61 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/108","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 11

Abstract

Let S S be a surface and let Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) be the mapping class group of S S permuting a Cantor subset K ⊂ S K \subset S . We prove two structure theorems for normal subgroups of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) .

(Purity:) if S S has finite type, every normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) either contains the kernel of the forgetful map to the mapping class group of S S , or it is ‘pure’ — i.e. it fixes the Cantor set pointwise.

(Inertia:) for any n n element subset Q Q of the Cantor set, there is a forgetful map from the pure subgroup PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) to the mapping class group of ( S , Q ) (S,Q) fixing Q Q pointwise. If N N is a normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) contained in PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) , its image N Q N_Q is likewise normal. We characterize exactly which finite-type normal subgroups N Q N_Q arise this way.

Several applications and numerous examples are also given.

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大映射类群的正规子群

设S S是一个曲面，设Mod (S,K) \operatorname {Mod}(S,K)是S S置换康托尔子集K的映射类群;S K \子集S。我们证明了Mod (S,K) \operatorname {Mod}(S,K)的正规子群的两个结构定理。(纯度:)如果S S具有有限类型，则Mod (S,K) \operatorname {Mod}(S,K)的每个正规子群要么包含到S S的映射类群的遗忘映射的核，要么是'纯' -(惯性:)对于Cantor集合的任意n个元素子集Q Q，存在一个从纯子群PMod (S,K) \operatorname {Mod}(S,K)到(S,Q) (S,Q)的映射类群(S,Q) (S,Q)的映射类群(S,Q)的映射，并定点地固定Q Q。如果N N是包含在PMod (S,K) \operatorname {PMod}(S,K)中的Mod (S,K) \operatorname {PMod}(S,K)的正规子群，那么它的像nqn_q同样是正规的。我们精确地描述了哪些有限型正规子群nqn_q是这样产生的。并给出了几个应用和大量的例子。

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