Abstract: Geometric structures are ubiquitous objects in low dimensional topology. Roughly speaking, such structures allow one to transport geometry from some nice model space onto a manifold by using charts with appropriate transition functions. Common examples include familiar objects such as Euclidean and hyperbolic structures as well as more exotic structures like affine and Anti-de Sitter structures. Despite their importance, in practice it can be hard to construct examples because the space of all possible charts is very large. In this context, the goal of this talk is to explain the philosophy that gluing equations can be viewed as a growing class of tools designed to “discretize†the problem of constructing geometric structures by using the combinatorial data of a triangulation to shrink the set of possible charts to a more manageable space (i.e. finite dimensional). This philosophy will be motivated by examples including Thurston’s original gluing equations for hyperbolic structures and more recent gluing equations of myself and Casella for projective structures. This work is joint with Alex Casella.
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