Volumes of SLn(ℂ)–representations of hyperbolic 3–manifolds

Pitsch, Wolfgang¹ ; Porti, Joan¹

¹ BGSMath and Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Spain

Geometry & topology, Tome 22 (2018) no. 7, pp. 4067-4112.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.4067

Classification : 14D20, 57M50, 57R20, 57T10
Keywords: volume, hyperbolic manifold, characteristic class, representation variety, Schäfli formula, flat bundle

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