#### Volume 22, issue 7 (2018)

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Volumes of $\mathrm{SL}_n(\mathbb{C})$–representations of hyperbolic $3$–manifolds

### Wolfgang Pitsch and Joan Porti

Geometry & Topology 22 (2018) 4067–4112
##### Abstract

Let $M$ 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 ${\pi }_{1}\left(M\right)$ in ${SL}_{n}\left(ℂ\right)$. Our proof follows the strategy of Reznikov’s rigidity when $M$ is closed; in particular, we use Fuks’s approach to variations by means of Lie algebra cohomology. When $n=2$, 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.

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##### Keywords
volume, hyperbolic manifold, characteristic class, representation variety, Schäfli formula, flat bundle
##### Mathematical Subject Classification 2010
Primary: 14D20, 57M50
Secondary: 57R20, 57T10