Tohoku Mathematical Journal
2021
December
SECOND SERIES VOL. 73, NO. 4

Tohoku Math. J.
73 (2021), 597-626

Title INFINITESIMAL GLUING EQUATIONS AND THE ADJOINT HYPERBOLIC REIDEMEISTER TORSION

Author Rafał Siejakowski

(Received October 29, 2019, revised August 28, 2020)
Abstract. We establish a link between the 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. This provides 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 verify the generalized conjecture in the case of the sister manifold of the figure-eight knot complement.

Mathematics Subject Classification. Primary 57Q10; Secondary 57M50.
Key words and phrases. Hyperbolic 3-manifolds, Reidemeister torsion, ideal triangulations, gluing equations.

To the top of this page

Back to the Contents