We study a particular distinguished component (the 'Hitchin component') of the space of surface group representations to SL(3,\R). In this setting, both Hitchin (via Higgs bundles) and the more ancient subject of affine spheres associate a bundle of holomorphic differentials over Teichmuller space to this component of the character variety. We focus on a ray of holomorphic differentials and provide a formula, tropical in appearance, for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how t
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