Abelian tropical covers

For a Riemann surface (and really all “nice” topological spaces), all abelian covering spaces are classified by the singular cohomology groups. Similarly, for an algebraic curve, all finite abelian coverings spaces are classified by the étale cohomology groups, and for number fields the abelian extension are classified in terms of class field theory. We consider the analogous question for tropical curves: Given a tropical curve, how can you classify all of its (finite) abelian covers?

This question is less trivial than it first appears, since, due to the presence of dilation, there are many unramified tropical covers that are not covering spaces in a topological sense. To resolve this probelm, we propose a so-called dilated cohomology group that classifies all abelian covers with a certain fixed dilation profile. In this talk I will give an elementary introduction to this cohomology theory and highlight its connections to the Bass-Serre theory of graphs of groups. I will also present a framework to study the realizability problem for abelian tropical covers, which, in the case of cyclic covers, is surprisingly closely related to the nowhere zero flow problem on a finite graph. This is based on joint work with Yoav Len and Dmitry Zakharov.