## Abstract

Let *Γ* be a tropical curve (or metric graph), and fix a base point *p*∈*Γ*. We define the Jacobian group *J*(*G*) of a finite weighted graph *G*, and show that the Jacobian *J*(*Γ*) is canonically isomorphic to the direct limit of *J*(*G*) over all weighted graph models *G* for *Γ*. This result is useful for reducing certain questions about the Abel–Jacobi map *Φ*
_{p}:*Γ*→*J*(*Γ*), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that *J*(*G*) is finite if and only if the edges in each 2-connected component of *G* are commensurable over ℚ. As an application of our direct limit theorem, we derive some local comparison formulas between *ρ* and \({\varPhi}_{p}^{*}(\rho)\) for three different natural “metrics” *ρ* on *J*(*Γ*). One of these formulas implies that *Φ*
_{p} is a tropical isometry when *Γ* is 2-edge-connected. Another shows that the canonical measure *μ*
_{Zh } on a metric graph *Γ*, defined by S. Zhang, measures lengths on *Φ*
_{p}(*Γ*) with respect to the “sup-norm” on *J*(*Γ*).

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Baker, M., Faber, X. Metric properties of the tropical Abel–Jacobi map.
*J Algebr Comb* **33**, 349–381 (2011). https://doi.org/10.1007/s10801-010-0247-3

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DOI: https://doi.org/10.1007/s10801-010-0247-3