Algorithmic approaches to Teichmüller curves

Teichmüller curves are algebraic curves defined over number fields in moduli space  $M_g$ of closed Riemann surfaces of genus g. They are obtained from special translation surfaces by a surprisingly explicit construction. An important class of such translation surfaces are the so-called origamis which are determined by purely combinatorial data. The aim of this  project is to investigate the following three aspects for  Teichmüller curves of special classes of origamis (and more generally of imprimitive translation surfaces) based on computer experiments: their boundary points in the Deligne-Mumford compactification, the deficiency of their Veech groups from being a congruence group and their analogs in moduli spaces of tropical curves.