On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds
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...
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| Format: | Theses and Dissertations |
| Language: | English |
| Published: |
2017
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| Online Access: | http://hdl.handle.net/10356/72448 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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. |
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