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Volumes of
$\mathrm{SL}_n(\mathbb{C})$–representations of hyperbolic
$3$–manifolds
Wolfgang Pitsch and Joan Porti |
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Geometry & Topology 22 (2018) 4067–4112
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Abstract |
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Let be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of in . Our proof follows the strategy of Reznikov’s rigidity when is closed; in particular, we use Fuks’s approach to variations by means of Lie algebra cohomology. When , we get Hodgson’s formula for variation of volume on the space of hyperbolic Dehn fillings. Our formula also recovers the variation of volume on the space of decorated triangulations obtained by Bergeron, Falbel and Guilloux and Dimofte, Gabella and Goncharov. |
Keywords
volume, hyperbolic manifold, characteristic class,
representation variety, Schäfli formula, flat bundle
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Mathematical Subject Classification 2010
Primary: 14D20, 57M50
Secondary: 57R20, 57T10
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References |
Publication
Received: 19 April 2017
Revised: 21 March 2018
Accepted: 20 May 2018
Published: 6 December 2018
Proposed: Walter Neumann
Seconded: Ian Agol, Bruce Kleiner
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Authors |
| Wolfgang Pitsch | |
| BGSMath and Departament de
Matemàtiques Universitat Autònoma de Barcelona Bellaterra Spain |
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| http://mat.uab.es/~pitsch/ | |
| Joan Porti | |
| BGSMath and Departament de
Matemàtiques Universitat Autònoma de Barcelona Bellaterra Spain |
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| http://mat.uab.cat/~porti/ | |
