Algebraic and geometric solutions of hyperbolicity equations
This paper studies differences between algebraic and geometric solutions of hyperbolicity equations for ideally triangulated manifolds. More precisely, let \(M\) be a compact three manifold with boundary a union of tori. Consider an ideal triangulation \(\Delta_1,\ldots,\Delta_n\) of the interior of \(M\), i.e. the \(\Delta_j\) have no vertices. Ideal hyperbolic tetrahedra are parametrized by a complex number \(z\in\mathbb C\) with positive imaginary part (if the imaginary part is zero or negative, then the tetraedron is flat or negatively oriented, respectively). Thus consider \(z_1,\ldots,z_n\in\mathbb C\), so that \(z_i\) describes a hyperbolic structure on \(\Delta_i\). Those structures on the \(\Delta_j\) are always compatible along the 2-dimensional faces, but not along the edges, and they are not necessarily complete. Compatibility and completeness are read algebraically on the \(z_j\), generating some equations (a compatibility equation for each edge and a completeness equation for each end of the interior of \(M\)). When a solution \(z_1,\ldots,z_n\) satisfies the compatibility and completeness equations, and the imaginary part of each \(z_j\) is positive, then it induces a hyperbolic metric on \(M\). Any algebraic solution inducing a structure is called geometric. In addition, the compatibility equations are not only polynomial, but they concern also the arguments of the \(z_j\). Sometimes, the completeness equations are replaced by Dehn filing ones, so that the geometric structure is not complete, and its completion is a Dehn filling of \(M\), as in the proof of Thurston's hyperbolic Dehn filling [cf. \textit{W. D. Neumann} and \textit{D. Zagier}, Topology 24, 307--332 (1985; Zbl 0589.57015)]. Algebraic solutions and their difference with geometric solutions are analyzed in this paper. In particular it is shown that the subset of geometric solutions is open in the set of algebraic ones. Also uniqueness of the geometric solutions satisfying hyperbolic Dehn filling equations is proved. Interesting examples are constructed at the end of the paper, especially a non-hyperbolic (containing tori) manifold admiting a partially flat solution of the compatibility and completeness equations.
scientific article; zbMATH DE number 2136834
scientific article