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

TL;DR

This paper explores the geometry of limits of positive Hitchin representations of surface groups, using positivity properties to describe apartments in the associated Euclidean buildings, advancing understanding of boundary points in the Hitchin component.

Contribution

It provides an explicit geometric description of apartments in the limiting Euclidean buildings for Hitchin representations, leveraging positivity properties introduced by Fock and Goncharov.

Findings

Explicit description of apartments in the limiting building.

Connection between positivity properties and geometric limits.

Enhanced understanding of boundary points in Hitchin components.

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.