Positive configurations of flags in a building and limits of positive representations

Abstract

Parreau compactified the Hitchin component of a closed surface S of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of $$\pi _1(S)$$ on an $${\mathbb {R}}$$ -Euclidean building. In this paper, we use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building.