Certifying hyperbolicity of fibred 3-manifolds

Thesis

In this thesis, we study the algorithmic problem of deciding whether a 3-manifold fibres over the circle, and if so, whether it is hyperbolic. We are not only concerned with decidability of these questions, but also with their computational complexity: a 3-manifold is described by a triangulation, and the efficiency of an algorithm is measured against the number of tetrahedra in the triangulation.

We prove that the problem of deciding whether a triangulated orientable 3-manifold fibres over the circle lies in the complexity class NP, generalising a result of Schleimer that only applied to atoroidal 3-manifolds. By design, our certificate for fibredness can be used to recover the monodromy of a fibration. Building on our previous work on algorithmic Nielsen-Thurston classification of surface mapping classes, we can decide whether the monodromy is pseudo-Anosov, and thus whether the fibred 3-manifold is hyperbolic. More precisely, we show that hyperbolicity of a triangulated orientable fibred 3-manifold can be certified in polynomial time in the number of tetrahedra in the triangulation and the Euler characteristic of a fibre. In the special case of knots in the 3-sphere, where the input is given as a planar diagram, our result implies that the problem of deciding whether a fibred knot is hyperbolic lies in NP.

Authors: Baroni, F

• DOI: 10.5287/ora-2zzmdmemx
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: fibred 3-manifolds; hyperbolic 3-manifolds; 3-manifolds; algorithms
• Subjects: Mathematics