Let p be a prime number. The moduli space M_{g, Z/pZ} parametrizes isomorphism classes of étale Z/pZ-Galois covers of smooth curves of genus g. Let M_g denote the moduli space of smooth curves of genus g. Then M_{g, Z/pZ} is itself an étale covering of M_g of degree p^{2g} - 1.

We study an associated moduli space of unramified tropical Z/pZ-covers of curves of genus g. Its rational homology is conjecturally identified with the top weight cohomology of M_{g, Z/pZ}. When g > 2, we identify a nested sequence of contractible loci in the moduli space of tropical covers and show that it is simply connected. In the case g=2, we also determine the homotopy type of the tropical space, showing that it is contractible for small p and is a wedge of spheres for p >3. This is joint work with Yassine El Maazouz, Paul Alexander Helminck, Felix Röhrle, and Pedro Souza.