The Homeomorphism Problem For Hyperbolic Manifolds I

We give a bounded runtime solution to the homeomorphism problem for closed hyperbolic 3-manifolds. This is an algorithm which, given two triangulations of hyperbolic 3-manifolds by at most $t$ tetrahedra, decides if they represent the same hyperbolic 3-manifold with runtime bounded by $$2^{2^{t^{O(t)}}}.$$ We do this by first finding a hyperbolic structure on each manifold given as a geometric triangulation and then comparing the two as geometric manifolds.