Elements of the tropical vertex group are formal families of
symplectomorphisms of the 2-dimensional algebraic torus. Commutators in the
group are related to Euler characteristics of the moduli spaces of quiver
representations and the Gromov-Witten theory of toric surfaces. After a short
survey of the subject (based on lectures of Pandharipande at the 2009 Geometry
summer school in Lisbon), we prove new results about the rays and symmetries of
scattering diagrams of commutators (including previous conjectures by
Gross-Siebert and Kontsevich). Where possible, we present both the quiver and
Gromov-Witten perspectives.Comment: 43 page