Ideal triangulations and geometric transitions

TL;DR

This paper extends Thurston's ideal triangulation method to study continuous geometric transitions from hyperbolic to anti de Sitter structures, revealing new solution components and rigidity properties.

Contribution

It generalizes Thurston's gluing equations to pseudo-complex algebras, identifying continuous families of hyperbolic and AdS structures in punctured torus bundles.

Findings

Identified two components of real solutions for hyperbolic to AdS transition.

Constructed continuous families of real projective structures connecting hyperbolic and AdS geometries.

Analyzed rigidity properties of AdS structures with tachyon singularities.

Abstract

Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to anti de Sitter (AdS) geometry. Our approach involves solving Thurston's gluing equations over several different shape parameter algebras. In the case of a punctured torus bundle with Anosov monodromy, we identify two components of real solutions for which there are always nearby positively oriented solutions over both the complex and pseudo-complex numbers. These complex and pseudo-complex solutions define hyperbolic and AdS structures that, after coordinate change in the projective model, may be arranged into one continuous family of real projective structures. We also study the rigidity properties of certain AdS structures with tachyon singularities.