Georgia Institute of Technology College of Sciences

Given a graph X and a group G, a G-cover of X is a morphism of graphs X’ --> X together with an invariant G-action on X’ that acts freely and transitively on the fibers. G-covers are classified by their monodromy representations, and if G is a finite abelian group, then the set of G-covers of X is in natural bijection with the first simplicial cohomology group H1(X,G).

In tropical geometry, we are naturally led to consider more general objects: morphisms of graphs X’ --> X admitting an invariant G-action on X’, such that the induced action on the fibers is transitive, but not necessarily free. A natural question is to classify all such covers of a given graph X. I will show that when G is a finite abelian group, a G-cover of a graph X is naturally determined by two data: a stratification S of X by subgroups of G, and an element of a cohomology group H1(X,S) generalizing the simplicial cohomology group H1(X,G). This classification can be viewed as a tropical version of geometric class field theory, and as an abelianization of Bass--Serre theory.

I will discuss the realizability problem for tropical abelian covers, and the relationship between cyclic covers of a tropical curve C and the corresponding torsion subgroup of Jac(C). The realizability problem for cyclic covers of prime degree turns out to be related to the classical nowhere-zero flow problem in graph theory.

Joint work with Yoav Len and Martin Ulirsch.