  We establish a link between the holomorphic derivatives of Thurston's
hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic
3-manifold and the cohomology of the sheaf of infinitesimal isometries.
Moreover, we provide a geometric reformulation of the non-abelian Reidemeister
torsion corresponding to the adjoint of the monodromy representation of the
hyperbolic structure. These results are then applied to the study of the
`1-loop Conjecture' of Dimofte-Garoufalidis, which we generalize to arbitrary
1-cusped hyperbolic 3-manifolds. We rigorously verify the generalized
conjecture in the case of the sister manifold of the figure-eight knot
complement.
