We prove that many normal subgroups of the extended mapping class group of a surface with punctures are geometric, that is, that their automorphism groups and abstract commensurator groups are isomorphic to the extended mapping class group. In order to apply our theorem to a normal subgroup we require that the "minimal supports" of its elements satisfy a certain complexity condition that is easy to check in practice. The key ingredient is proving that the automorphism groups of many simplicial complexes associated to punctured surfaces are isomorphic to the extended mapping class group. This resolves many cases of a metaconjecture of N. V. Ivanov and extends work of Brendle-Margalit, who prove the result for closed surfaces.

The author would like to thank his supervisor, Tara Brendle, for her helpful guidance and support. He is grateful to Dan Margalit for several helpful discussions and suggestions that greatly improved the paper. He would also like to thank Javier Aramayona, Vaibhav Gadre, Tyrone Ghaswala, Chris Leininger, Johanna Mangahas, and Shane Scott for their support and helpful discussions about the paper. Finally, he thanks the referee for suggestions that improved the clarity of the arguments. This work was partially supported by the National Research Fund, Luxembourg.