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.

26 pages, 4 figures, Minor corrections (typos, etc.)