On a bounded runtime algorithm for the homeomorphism problem for hyperbolic manifolds

Thesis

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 2²tO(t).

Along the way we prove two supplementary results, both of which we also prove in higher dimensions. First, we prove that one can compare geometric triangulations of hyperbolic n-manifolds in time bounded by 2²tO(nt³), n≥ 4, 2²tO(t), n = 3.

Second, we show that the length R of a systole of a closed or finite hyperbolic n-manifold (n ≥ 3) admitting a triangulation by t n-simplices can be bounded below by a function of n and t, namely R ≥ 1/ 2(nt)o(n⁴t).

Authors: Scull, J

• DOI: 10.5287/ora-g2avpoked
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English