Translation surfaces with poles and their moduli spaces

On a Riemann surface $X$, every meromorphic abelian differential determines a translation surface and a representation $\chi\colon\pi_1(X)\to \mathbb C$ called period character. In this seminar we shall talk about strata of meromorphic differential with prescribed zeros and poles. Strata admit a natural foliation in which every leaf comprises translation surfaces with the same period character. In the special case of H(1,1;-2) each leaf is homeomorphic to the Loch Ness monster and it carries itself a transition structure.