Solving Thurston's equation in a commutative ring (Q2797315)

Thurston's equations are a collection of compatibility criteria for assembling hyperbolic ideal tetrahedra into a hyperbolic 3-manifold. Various versions of these equations have been useful for understanding the relationship between the geometry and topology of 3-manifolds. For instance, they have been used to study the limiting behavior of hyperbolic volume under Dehn filling [\textit{W. D. Neumann} and \textit{D. Zagier}, Topology 24, 307--332 (1985; Zbl 0589.57015)] and to show that if a triangulated 3-manifold has a solution to the equations, then every edge of the triangulation is essential in the manifold [\textit{H. Segerman} and \textit{S. Tillmann}, Contemp. Math. 560, 85--102 (2011; Zbl 1333.57017)].NEWLINENEWLINEIn the paper under review, the author considers the topological significance of solutions to Thurston's equations in an arbitrary commutative ring with identity. The main result concerns a connected, triangulated 3-dimensional pseudo-manifold \(M\) without boundary and a commutative ring \(R\) with identity such that Thurston's equations are solvable in \(R\). The paper shows that each edge of the triangulation lifts to an edge in the universal cover with distinct endpoints. Furthermore, if \(M\) is a closed, connected 3-manifold and if an edge of the triangulation is a loop, then there is a representation of \(\pi_1(M)\) into \(\mathrm{PGL}(2,R)\) sending the element of \(\pi_1(M)\) corresponding to the loop to a non-identity element. When \(R\) is the complex numbers, the theorem was first proved by Segerman and Tillmann [loc. cit.].NEWLINENEWLINETo prove the main theorem, the author introduces the Homogenous Thurston's Equations (HTE) for any ring. If the ring is a commutative ring with identity, then solutions of the HTE in the group of invertible elements of the ring correspond to solutions to Thurston's equations in \(R\). Pseudo-developing maps and holonomy representations are constructed using solutions to HTE and cross ratios in \(R^2\).NEWLINENEWLINEThe paper also contains a few other interesting results. The author shows that solutions to HTE correspond to critical points of a Boltzman entropy function and suggests that this may be used to construct invariants of 3-manifolds. The author also creates a universal construction for a triangulated 3-dimensional pseudo-manifold, which he calls the Thurston Ring. Thurston's equations have a solution in \(R\), a commutative ring with identity, if and only if there is a non-trivial ring homomorphism from the Thurston ring to \(R\).NEWLINENEWLINEThe final section of the paper provides a collection of examples relating solutions to Thurston's equations in finite rings to the combinatorics of edges in the triangulation, including a new proof of a result of Rubinstein and Tillmann.