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A plane tropical curve $\Gamma$ is the corner locus of a tropical polynomial in two variables. That is, it is the set of points at which the tropical polynomial, which is a piecewise-linear concave function, fails to be linear. Equivalently, $\Gamma$ is a weighted graph in the plane satisfying certain, fairly rigid conditions. (For instance, each edge has rational slope, and the weighted sum of the primitive vectors of the edges around a given vertex is zero. See Gathmann for a more thorough introduction to tropical plane curves.) Of particular importance to my question is the fact that $\Gamma$ has no finite points of valence 1.

When the actual polynomial defining $\Gamma$ is not of particular importance to the question at hand, we can shift our attention to more abstract structures. Such an approach was recently used successfully by Joyner, Ksir, and Melles. They first define a star-shaped set to be any set of the form $$ S(n,r) = \{ z \in \mathbb{C} \mid z = t \exp(2\pi i/n) \ \text{for some $t \in [0,r)$ and $k \in \mathbb{Z}$}\}, $$ where $n$ is a positive integer and $r$ is a positive real number. The set $S(n,r)$ is given the path metric and the metric topology. An abstract tropical curve is then defined to be a compact connected topological space with the property that every point has a neighborhood homeomorphic and isometric to a star-shaped set. Further, informally speaking, each edge is given a positive integer weight and no finite leaves are allowed.

By shifting our focus to abstract tropical curves, we lose the rigid constraints on tropical plane curves, while we retain their topological and metric structure. Moreover, this approach is more general: we are no longer restricted to plane curves, and can consider tropical curves in larger-dimensional spaces.


Let $\Gamma$ be a tropical plane curve, and let $G$ be a nontrivial subgroup of $\text{Aut}(\Gamma)$. Under what conditions is the quotient $\Gamma/G$ is also a tropical curve. Because I fear this may be too restrictive a question---I see no obvious examples---I would like to include abstract tropical curves in my question. If $\Gamma$ is an (abstract) tropical curve, when is $\Gamma/G$ an abstract tropical curve? In this case, it is relatively easy to cook up examples of genus zero abstract tropical curves for which this works. A rotationally symmetric genus one curve also works, with $G$ the group of rotations. However, it fails in other cases. For instance, if $G$ contains a reflection, it is possible that $\Gamma/G$ has finite leaves. Can anyone supply a more illuminating example?

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A good place to start would be to look at covering spaces. A covering space of a topological space X is a space that is locally isomorphic to X but globally unwinds some of the topology. A good reference is Hatcher's Algebraic Topology notes (section 1.3 has what you'd need in the first couple of pages). Sometimes X is the quotient of its covering by a group action. This property is called normality and there is a nice group theoretic condition for it. As a fun exercise, before you even begin, try to find some graphs Y with group actions such that Y/G is the figure 8.

In a certain sense, covering spaces are the "easiest" types of quotients. This is because the action is properly discontinuous. For this reason, the quotient inherits the tropical structure of the original graph. For more exotic group actions, you may want to look at Matthias Herold's preprint: "Tropical orbit spaces and the moduli spaces of elliptic tropical curves." There should be some clues in that paper for how to work with graphs with finite leaves.

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