  We show that if $X$ is a smooth quasi-projective $3$-fold admitting a
flopping contraction, then the fundamental group of an associated simplicial
hyperplane arrangement acts faithfully on the derived category of $X$. The main
technical advance is to use torsion pairs as an efficient mechanism to track
various objects under iterations of the flop functor (respectively, mutation
functor). This allows us to relate compositions of the flop functor
(respectively, mutation functor) to the theory of Deligne normal form, and to
give a criterion for when a finite composition of $3$-fold flops can be
understood as a tilt at a single torsion pair. We also use this technique to
give a simplified proof of the result of Brav-Thomas for Kleinian
singularities.
