Lengths are coordinates for convex structures (Q2497078)

Associated to a hyperbolic 3-manifold \(N\) is a collection of geometric invariants. It is an important problem to decide which subset of invariants determines \(N\). In the paper under review, the authors consider the case where \(N\) is a geometrically finite orientable hyperbolic 3-manifold, and consider the space \({\mathcal P}(N, \underline{\alpha})\) of all geometrically finite hyperbolic structures on \(N\) such that the convex core (with finite, non-zero volume) is bent along a set \(\underline{\alpha}:=\{\alpha_1, \ldots,\alpha_n\}\) of simple closed curves. The main theorem in the paper is that the map \(L :{\mathcal P}(N, \underline{\alpha}) \rightarrow {\mathbb{R}}^n\) which associates to each structure \(\sigma\) the hyperbolic lengths \((l_{\alpha_1}(\sigma), \ldots, l_{\alpha_1}(\sigma))\) is an injective local diffeomorphism. (In fact, they claim that they can show that \(L\) is actually a diffeomorphism onto its image). This means in particular that \(\sigma \in {\mathcal P}(N, \underline{\alpha})\) is parametrized by the lengths of the curves in \(\alpha\), which is a length analogue of a result of \textit{F. Bonahon} and \textit{J.-P. Otal} [Ann. Math. (2) 160, No.3, 1013--1055 (2005; Zbl 1083.57023)] which states that the corresponding map \(\Theta : {\mathcal P}(N, \underline{\alpha}) \rightarrow {\mathbb{R}}^n\) associating to each structure \(\sigma\) the set of bending angles \((\theta_{\alpha_1}(\sigma), \ldots, \theta_{\alpha_1}(\sigma))\) is a diffeomorphism onto a convex subset of \((0,\pi]^n\). In fact, the authors prove the stronger result that any mixed parametrization using either the bending angle or the length of \(\alpha_i \in \underline{\alpha}\), for \(i=1, \ldots, n\) is an injective local diffeomorphism on \({\mathcal P}(N, \underline{\alpha})\), from which the main theorem follows. The result generalizes substantially previous results of the second author with \textit{L. Keen} [Topology 43(2), 447--491 (2004; Zbl 1134.30030)], as well as with \textit{R. Diaz} [Trans. Am. Math. Soc 356(2), 621--658 (2004; Zbl 1088.30043)]. The authors first double the convex core across the boundary, to obtain a cone hyperbolic 3-manifold, the main tool used in this paper is then the local deformation theory of cone manifolds developed by Hodgson and Kerkhoff. These methods differ somewhat from the techniques used in these two previous papers, which are more elementary and direct.