We use geometric methods to show that, given any 3-manifold M, and g a sufficiently large integer, the mapping class group Mod(Sigma(g,1)) contains a coset of an abelian subgroup of rank inverted left perpendicular g/2 inverted right perpendicular, consisting of pseudo-Anosov monodromies of open-book decompositions in M. We prove a similar result for rank two free cosets of Mod(Sigma(g,1)). These results have applications to a conjecture of Andersen, Masbaum, and Ueno about quantum representations of surface mapping class groups. For surfaces with boundary, and large enough genus, we construct cosets of abelian and free subgroups of their mapping class groups consisting of elements that satisfy the conjecture. The mapping tori of these elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viro invariants volume conjecture.