I am interested in the mapping class groups on surfaces.
The most common elements in the mapping class groups are so called pseudo-Anosovs. I try to understand which pseudo-Anosovs are the most simplest in the mapping class groups. I describe my goal more clearly. Pseudo-Anosovs possess many complicated (and beautiful) properties from the view points of the dynamical systems and the hyperbolic geometry. There are some quantities which reflect those complexities of pseudo-Anosovs. Entropies and volumes (i.e, volumes of mapping tori) are examples. We fix the topological type of the surface and we consider the set of entropies (the set of volumes) coming from the pseudo-Anosov elements on the surface. Then one can see that there exists a minimum of the set. That is, we can talk about the pseudo-Anosovs with the minimal entropies (pseudo-Anosovs with the minimal volumes). I would like to know which pseudo-Anosov achieves the minimal entropy/ with the minimal volume.
Recently, Gabai, Meyerhoff and Milley determined hyperbolic closed 3-manifolds
and hyperbolic 3-manifolds with one cusp with very small volume. Intriguingly, the result implies that those hyperbolic 3-manifolds are obtained from the single hyperbolic 3-manifold by Dehn filling. Some experts call the single 3-manifold the ``magic manifold". Said differently, the magic manifold is a parent manifold of the hyperbolic manifolds with very small volume.
It seems likely that we have the same story in the world of pseudo-Anosovs with the very small entropies. This conjecture is based on the recent works of myself and other specialists. We note that there are infinitely many topological types of surfaces.
(for example, the family of closed orientable surface with genus g). For the mapping class group of each surface we know that there exists a pseudo-Anosov element with the minimal entropy. Thus, of course, there exist infinitely many pseudo-Anosov elements with the minimal entropies. It might be true that all minimizers are obtained from the magic 3-manifold. When I work on this project, I sometimes think of our universe.