On deformations of hyperbolic 3–manifolds with geodesic boundary

Abstract

Let $M$ be a complete finite-volume hyperbolic 3–manifold with compact non-empty geodesic boundary and $k$ toric cusps, and let $\mathcal{T}$ be a geometric partially truncated triangulation of $M$. We show that the variety of solutions of consistency equations for $\mathcal{T}$ is a smooth manifold or real dimension $2k$ near the point representing the unique complete structure on $M$. As a consequence, the relation between deformations of triangulations and deformations of representations is completely understood, at least in a neighbourhood of the complete structure. This allows us to prove, for example, that small deformations of the complete triangulation affect the compact tetrahedra and the hyperbolic structure on the geodesic boundary only at the second order.