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

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.