Volumes of $\mathrm{SL}_n\mathbb{C}$-representations of fundamental groups of hyperbolic 3-manifold groups

Wolfgang Pitsch, Joan Porti

Published 2017-04-05Version 1

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 $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows the strategy of Reznikov's rigidity when $M$ is closed, in particular we use Fuks' approach to variations by means of Lie algebra cohomology. When $n=2$, we get back Hodgson's formula for variation of volume on the space of hyperbolic Dehn fillings. Our formula also yields the variation of volume on the space of decorated triangulations obtained by Bergeron-Falbel-Guillou or Dimofte-Gabella-Goncharov.