Normal subgroups of mapping class groups and the metaconjecture of Ivanov
Normal subgroups of mapping class groups and the metaconjecture of Ivanov
复制标题
DOI：
10.1090/jams/927
复制
发表时间：
2017-10
影响因子：
3.9
通讯作者：
Tara E. Brendle;D. Margalit
中科院分区：
文献类型：
--
作者：
Tara E. Brendle;D. Margalit
证明了如果闭曲面的扩张映射类群的正规子群有一个支撑度足够小的元，则它的自同构群和抽象公度子群都与扩张映射类群同构。证明依赖于我们证明的另一个定理，即许多与闭曲面相关的单纯复形都有同构于扩张映射类群的自同构群。这些结果解决了N.V.Ivanov的亚猜想，该猜想断言对于一大类这样的对象，任何与曲面相关的“足够丰富的”对象都有同构于扩展映射类群的自同构群。作为应用，我们证明了：(1)直角Artin群和曲面群不能同构于包含小支撑元的映射类群的正规子群；(2)如果不同映射类群的正规子群都具有小支撑元，则它们不能同构；(3)具有小支撑元的映射类群的不同正规子群不同构。我们的结果还为映射类群的正规子群的分类提供了一个新的框架。
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support, then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N. V. Ivanov, which asserts that any “sufficiently rich” object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.