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

Wolfgang Pitsch; Joan Porti

doi:10.2140/gt.2018.22.4067

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arXiv · 1704.01321

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

Wolfgang Pitsch·Joan Porti

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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.

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BibTeX RIS

Wolfgang Pitsch, Joan Porti. 2018-03-21. Volumes of $\mathrm{SL}_n\mathbb{C}$-representations of hyperbolic 3-manifolds. https://doi.org/10.2140/gt.2018.22.4067

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