Infinitesimal gluing equations and the adjoint hyperbolic Reidemeister torsion

2017/10/05 by Rafał M. Siejakowski, Siejakowski, Rafał
Mathematics · Medicine · #57M50 (Secondary) #57Q10 (Primary) #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1710.02109

openalex publication_date 2017/10/05 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

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.

Related

• Adjoint twisted Reidemeister torsion and Gram matrices
• Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps
• The quantum content of the gluing equations
• Reidemeister torsion, Complex volume, and Zograf infinite product for hyperbolic 3-manifolds with cusps
• Twisted Alexander invariant and non-abelian Reidemeister torsion for hyperbolic three-dimensional manifolds with cusps