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Abstract Suppose that is a free product , where each of the groups is torsion‐free and is a free group of rank . Let be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of where a given element has bounded length to the ‐factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of as a hyperbolic group relative to the collection of subgroups together with a given nonperipheral cyclic subgroup. The main theorem is new even in the case that , in which case is the Culler–Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions. 

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. 

Let G be a free-by-cyclic group or a 2-dimensional right-angled Artin group. We provide an algebraic and a geometric characterization for when each aspherical simplicial complex with fundamental group isomorphic to G has minimal volume entropy equal to 0. In the nonvanishing case, we provide a positive lower bound to the minimal volume entropy of an aspherical simplicial complex of minimal dimension for these two classes of groups. Our results rely upon a criterion for the vanishing of the minimal volume entropy for 2-dimensional groups with uniform uniform exponential growth. This criterion is shown by analyzing the fiber π1-growth collapse and non-collapsing assumptions of Babenko-Sabourau.