  We use geometric methods to show that given any $3$-manifold $M$, and $g$ a
sufficiently large integer, the mapping class group
$\mathrm{Mod}(\Sigma_{g,1})$ contains a coset of an abelian subgroup of rank
$\lfloor \frac{g}{2}\rfloor,$ consisting of pseudo-Anosov monodromies of
open-book decompositions in $M.$ We prove a similar result for rank two free
cosets of $\mathrm{Mod}(\Sigma_{g,1}).$ 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.
