Infinitesimal gluing equations and the adjoint hyperbolic Reidemeister torsion

TL;DR

This paper links hyperbolic gluing equations with cohomology of infinitesimal isometries and reformulates the adjoint Reidemeister torsion, advancing understanding of hyperbolic 3-manifolds and verifying a key conjecture.

Contribution

It establishes a geometric reformulation of the adjoint Reidemeister torsion and generalizes the 1-loop Conjecture to all 1-cusped hyperbolic 3-manifolds.

Findings

Linked derivatives of gluing equations to cohomology of infinitesimal isometries.

Reformulated non-abelian Reidemeister torsion geometrically.

Verified the generalized 1-loop Conjecture for the sister of the figure-eight knot.

Abstract

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.