In his famous lecture notes [http://library.msri.org/nonmsri/gt3m], W.Thurston suggested an explicit construction of hyperbolic structures on (ideally triangulated) \(3\)-manifolds by assigning a cross-ratio parameter to each ideal hyperbolic tetrahedron and stating equations between the parameters (depending on the combinatorics of the triangulation) which guarantee that the tetrahedra fit together to a complete hyperbolic structure without singularities at the edges. The paper under review generalizes this technique to construct more general projective structures rather than only hyperbolic ones.
Let us start with some necessary technical notions. An incomplete flag in \({\mathbb P}^3{\mathbb R}\) consists of a point \(V\in{\mathbb P}^3{\mathbb R}\) and a hyperplane \(\eta\) containing \(V\). A tuple of flags is non-degenerate if \(V_i\) is not contained in \(\eta_j\) for any \(i\not=j\). A tetrahedron of flags is a non-degenerate ordered \(4\)-tuple of incomplete flags such that the four points are in general position and there is a tetrahedron in \({\mathbb P}^3{\mathbb R}\) with vertices the four points, whose interior is disjoint from the four hyperplanes. (According to the authors, one should think of a tetrahedron of flags as vertices of a projective tetrahedron together with a hyperplane through each vertex.) The authors show that projective equivalence classes of tetrahedra of flags are determined by twelve edge ratios and twelve triple ratios satisfying some consistency equations. The ratios define a semi-algebraic set homeomorphic to \({\mathbb R}^5_{>0}\).
Given a \(3\)-manifold \(M\) with a \(\pi_1M\)-invariant ideal triangulation \(\widetilde{\Delta}\) of its universal covering, its space of triangulations of flags is the set of pairs \((\Phi,\rho)\), where \(\rho\) is a representation of \(\pi_1M\) to \(PGL(4,{\mathbb R})\) and \(\Phi\) is a \(\rho\)-equivariant map from the vertices of \(\widetilde{\Delta}\) to the space of incomplete flags, such that the image of each ideal tetrahedron is a tetrahedron of flags and the images of adjacent tetrahedra agree at the common faces. The authors prove that every triangulation of flags can be developed to a projective structure, possibly branched along the edges of the triangulation.
For two tetrahedra of flags to be gluable, their parameters have to satisfy some face pairing equations. When those are satisfied, there is a one-parameter family of inequivalent gluings, analogous to shearing ideal triangles along a common edge in hyperbolic surface theory. The authors represent this one-parameter family in a symmetric way by six gluing parameters satisfying some gluing consistency equations. They show that the parameter space of pairs of glued tetrahedra of flags is homeomorphic to \({\mathbb R}^{10}_{>0}\).
While in hyperbolic geometry, Thurston’s equations express the fact that the angles of hyperbolic tetrahedra around a common edge should add up to a multiple of \(2\pi\), the corresponding edge gluing equations for obtaining branched projective structures are harder to define and require to introduce what the authors call the monodromy complex. This is done in Section 4.
Altogether the authors obtain a parametrisation of the space of triangulations of flags (and hence via the developing map of branched projective structures) which is compatible with Thurston’s parametrization of branched hyperbolic structures and with its counterparts in anti-de Sitter and halfpipe geometry due to J. Danciger [J. Topol. 7, No. 4, 1118–1154 (2014; Zbl 1308.57006)].
Concerning properly convex projective structures, the authors discuss how to decorate generalized cusps with flags and they prove that each generalized cusp can be decorated with incomplete flags in only finitely many ways.
Finally, the authors use the equations to explicitly construct one-parameter families of finite-volume properly convex projective structures on the figure eight knot complement and its sister manifold.