An algorithm to decide whether a 3-manifold admits a hyperbolic structure of finite volume

Stephan Tillmann Sydney University


The algorithm of the title takes as input a triangulation or an ideal triangulation of a 3-manifold M, and decides which of the following, mutually exclusive cases holds: (0) M contains an essential sphere or an essential torus; (1) M is a small Seifert fibered space; (2) M admits a complete hyperbolic structure of finite volume. The main ingredients are normal surface theory, Groebner bases and computation of the Lobachevsky function. I will also discuss how this algorithm can be used in an algorithmic solution to the homeomorphism problem for 3-manifolds. Part of this talk is based on joint work with Feng Luo (Rutgers) and Tian Yang (Rutgers).

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