Brendle, T. E. and Margalit, D. (2019) Normal subgroups of mapping class groups and the metaconjecture of Ivanov. Journal of the American Mathematical Society, 32, pp. 1009-1070. (doi: 10.1090/jams/927)
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Abstract
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.
| Item Type: | Articles |
|---|---|
| Status: | Published |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Brendle, Professor Tara |
| Authors: | Brendle, T. E., and Margalit, D. |
| College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
| Journal Name: | Journal of the American Mathematical Society |
| Publisher: | American Mathematical Society |
| ISSN: | 0894-0347 |
| ISSN (Online): | 1088-6834 |
| Published Online: | 27 August 2019 |
| Related URLs: |
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