Certifying hyperbolicity of fibred 3-manifolds

One of the fundamental goals of low-dimensional topology is understanding the structure of 3-dimensional objects, formally known as "3-manifolds". Pioneering work by Thurston in the 1980s, later built upon by many mathematicians and culminating in Perelman's proof of the geometrisation conjecture, revealed that all 3-manifolds decompose into simpler "geometric pieces". These fall into eight "geometries", the most complex and rich being "hyperbolic geometry". As a consequence of the geometrisation theorem, we now know that hyperbolic 3-manifolds can be recognised algorithmically -- in other words, there is an algorithm that takes as input a 3-manifold and decides whether it is hyperbolic or not. However, no algorithm is known that can decide this efficiently.
Marc Lackenby has recently shown that the problem of deciding hyperbolicity is in co-NP, meaning that if a 3-manifold is not hyperbolic, it admits a certificate that witnesses this, which can be verified in polynomial time. The complementary question, on the other hand, remains open: can we efficiently certify that a 3-manifold is hyperbolic?
Aim: The main aim of this project is to devise an algorithm to efficiently certify the hyperbolicity of a broad class of 3-manifolds, namely those that are "fibred".
Methodology: this project consists of two key steps. Thurston's hyperbolisation theorem states that hyperbolicity of a fibred 3-manifold can be determined using only 2-dimensional data: a surface (the "fibre") and a continuous function (the "monodromy"). Thus, to certify that a fibred 3-manifold is hyperbolic, we propose the following approach: (1) provide a certificate consisting of a fibre and its monodromy; (2) develop an algorithm to verify that this data satisfies Thurston's conditions.
Research area: This falls within the EPSRC research area "Geometry & Topology".
No companies or collaborators will be involved