# Volumes of $\mathrm{SL}_n\mathbb{C}$-representations of hyperbolic 3-manifolds

## TL;DR

This paper establishes a variation formula for the volume of hyperbolic 3-manifolds represented in $ ext{SL}_n( ext{C})$, extending previous results and applying Lie algebra cohomology techniques.

## Contribution

It generalizes Hodgson's volume variation formula to higher rank groups and connects it with decorated triangulations and Lie algebra cohomology methods.

## Key findings

- Derived a new volume variation formula for $ ext{SL}_n( ext{C})$-representations.
- Extended Hodgson's formula to higher dimensions.
- Linked volume variations with decorated triangulations.

## 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 $M$ 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 and Dimofte-Gabella-Goncharov.