The Nielsen-Thurston classification theorem states that there are three kinds of surface homeomorphisms up to homotopy: periodic, reducible, and pseudo-Anosov.

In the introductory part of the talk, we will investigate the differences between these three categories, focusing on the wide array of geometric, topological, and dynamical properties that set pseudo-Anosov mapping classes apart from the rest. To this end, we will also introduce the curve graph, a combinatorial object associated with a surface, and describe how the dynamics of the action of a mapping class on the curve graph can be used to detect pseudo-Anosovness.

In the second part of the talk, we will see how to turn this characterisation into an algorithm for deciding whether a given mapping class is pseudo-Anosov. The key tools will be the theory of train tracks and their connection to the curve graph developed by Masur and Minsky.