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.

中文翻译：

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映射类群的正规子群和伊万诺夫的元猜想

我们证明，如果一个封闭曲面的扩展映射类群的正规子群有一个足够小的支持度元素，那么它的自同构群和抽象公度群都与扩展映射类群同构。该证明依赖于我们证明的另一个定理，该定理指出许多与闭合曲面相关联的单纯复形具有与扩展映射类群同构的自同构群。这些结果解决了 NV Ivanov 的元猜想，该猜想断言任何与表面关联的“足够丰富”的对象都具有与扩展映射类组同构的自同构组，对于此类对象的广泛类。作为应用程序，我们展示：(1) 直角 Artin 群和面群不能同构于包含小支座元素的映射类群的正规子群，(2) 如果它们都具有小支座元素，则不同映射类群的正规子群不能同构，和(3)具有小支持度元素的映射类群的不同正规子群不是同构的。我们的结果还为映射类组的正常子组的分类提出了一个新的框架。