On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds

• Main Author: Siejakowski, Rafał, M.
• Other Authors: Andrew James Kricker
• Format: Theses and Dissertations
• Language: English
• Published: 2017
• Subjects: DRNTU::Science::Chemistry
• Online Access: http://hdl.handle.net/10356/72448
• Institution: Nanyang Technological University

The non-abelian Reidemeister torsion is a numerical invariant of cusped hyperbolic 3-manifolds defined by J. Porti (1997) in terms of the adjoint holonomy representation of the hyperbolic structure. We develop a geometric approach to the definition and computation of the torsion using infinitesimal isometries. For manifolds carrying positively oriented geometric ideal triangulations, we establish a fundamental relationship between the derivatives of Thurston's gluing equations and the cohomology of the sheaf of infinitesimal isometries. Using these results, we obtain a partial confirmation of the "1-loop Conjecture" of Dimofte and Garoufalidis (2013) which expresses the non-abelian torsion in terms of the combinatorics of the gluing equations. In this way, we reduce the Conjecture to a certain normalization property of the Reidemeister torsion of free groups.