Adjoint Reidemeister torsion from hyperbolic gluing equations

Abstract: Motivated by the study of asymptotics of quantum invariants, Dimofte and Garoufalidis introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic $3$-manifold. They proved that its constant term can be written in terms of Neumann-Zagier data and the complex shape parameters of ideal tetrahedra, and conjectured that it equals the adjoint twisted Reidemeister torsion. On the other hand, in the study of asymptotics of Turaev-Viro type invariants of cusped $ 3$-manifolds, the authors, together with Liu, Sun and Yang, found that the one-loop terms of their asymptotic expansions could be written in terms of Gram matrices and decorated edge lengths of ideal tetrahedra. In this paper, we prove the conjecture of Dimofte and Garoufalidis and relate the one-loop term appearing in the Turaev-Viro type invariant to the adjoint twisted Reidemeister torsion.