Automorphism groups on tropical curves: some cohomology calculations

Let X be an abstract tropical curve and let G be a finite subgroup of the automorphism group of X . Let D be a divisor on X whose equivalence class is G -invariant. We address the following question: is there always a divisor D ′ in the equivalence class of D which is G -invariant? Our main result is that the answer is “yes” for all abstract tropical curves. A key step in our proof is a tropical analogue of Hilbert’s Theorem 90.