  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 surfaces without punctures.
