A geometric criterion to be pseudo-Anosov
  We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions.   Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface, and we show that such a class is pseudo-Anosov if its geodesic representative is "wide" in some hyperbolic 3-manifold homeomorphic to the trivial interval bundle over S. 
Richard P. Kent IV, Christopher J. Leininger
math.GT math.GR