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We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara. [ABSTRACT FROM AUTHOR]

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Г) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Г). In addition, we identify a milder condition for a finitely generated subgroup of A(Г) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Г. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups. [ABSTRACT FROM AUTHOR]

Given any generating set of any subgroup G of the mapping class group of a surface, we find an element f with word length bounded by a constant K depending only on the surface, and with the property that the minimal subsurface supporting a power of f is as large as possible for elements of G. In particular, if G contains a pseudo-Anosov map, we find one of word length at most K. We also find new examples of convex cocompact free subgroups of the mapping class group. [ABSTRACT FROM AUTHOR]