## Normal subgroups of mapping class groups and the metaconjecture of Ivanov

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Tara E. Brendle and Dan Margalit
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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.## References

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## Additional Information

**Tara E. Brendle**- Affiliation: School of Mathematics & Statistics, University Place, University of Glasgow, G12 8SQ, United Kingdom
- MR Author ID: 683339
- Email: tara.brendle@glasgow.ac.uk
**Dan Margalit**- Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332
- MR Author ID: 706322
- Email: margalit@math.gatech.edu
- Received by editor(s): May 9, 2018
- Received by editor(s) in revised form: April 10, 2019
- Published electronically: August 27, 2019
- Additional Notes: This material is based upon work supported by the EPSRC under grant EP/J019593/1 and the National Science Foundation under Grant Nos. DMS - 1057874 and DMS - 1510556.
- © Copyright 2019 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**32**(2019), 1009-1070 - MSC (2010): Primary 20F36; Secondary 57M07
- DOI: https://doi.org/10.1090/jams/927
- MathSciNet review: 4013739