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: | Thesis |
| Language: | English |
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2017
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| Online Access: | http://hdl.handle.net/10356/72448 |
| _version_ | 1826129823245795328 |
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| author | Siejakowski, Rafał, M. |
| author2 | Andrew James Kricker |
| author_facet | Andrew James Kricker Siejakowski, Rafał, M. |
| author_sort | Siejakowski, Rafał, M. |
| collection | NTU |
| description | 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. |
| first_indexed | 2024-10-01T07:46:49Z |
| format | Thesis |
| id | ntu-10356/72448 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T07:46:49Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | ntu-10356/724482023-02-28T23:55:24Z On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds Siejakowski, Rafał, M. Andrew James Kricker School of Physical and Mathematical Sciences DRNTU::Science::Chemistry 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. Doctor of Philosophy (SPMS) 2017-07-18T07:39:17Z 2017-07-18T07:39:17Z 2017 Thesis Siejakowski, R. M. (2017). On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds. Doctoral thesis, Nanyang Technological University, Singapore. http://hdl.handle.net/10356/72448 10.32657/10356/72448 en 123 p. application/pdf |
| spellingShingle | DRNTU::Science::Chemistry Siejakowski, Rafał, M. On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds |
| title | On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds |
| title_full | On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds |
| title_fullStr | On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds |
| title_full_unstemmed | On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds |
| title_short | On the geometric meaning of the non-abelian Reidemeister torsion of cusped hyperbolic 3-manifolds |
| title_sort | on the geometric meaning of the non abelian reidemeister torsion of cusped hyperbolic 3 manifolds |
| topic | DRNTU::Science::Chemistry |
| url | http://hdl.handle.net/10356/72448 |
| work_keys_str_mv | AT siejakowskirafałm onthegeometricmeaningofthenonabelianreidemeistertorsionofcuspedhyperbolic3manifolds |