Recognising pseudo-Anosov homeomorphisms

The Nielsen-Thurston classification identifies three types of elements in the mapping class group of a surface: periodic, reducible, and pseudo-Anosov. How does one concretely decide, given a mapping class, to which of the three categories it belongs? And can it be done efficiently?

We will present a new algorithm to solve this problem that runs in polynomial time in the word length of the mapping class and in the Euler characteristic of the surface. The techniques used are combinatorial, rooted in Masur and Minsky's characterisation of pseudo-Anosov mapping classes as those acting loxodromically on the curve graph of the surface.

We will also hint at how this algorithm can be applied to efficiently certify whether a given 3-manifold fibring over the circle is hyperbolic.