Sequences of Hitchin representations of Tree-type

TL;DR

This paper establishes conditions under which sequences of Hitchin representations converge to actions on trees in the Parreau boundary, using Fock-Goncharov coordinates to describe these limits.

Contribution

It provides new sufficient conditions for Hitchin representation sequences to have tree actions as limits, advancing understanding of their boundary behavior.

Findings

Conditions for convergence to tree actions in the Parreau boundary

Use of Fock-Goncharov coordinates to characterize limits

Extension of boundary analysis beyond Thurston compactification

Abstract

The Hitchin component is a connected component of the character variety of reductive group homomorphisms from the fundamental group of a closed surface S of genus greater than 1 to the Lie group PSL_m(R). The Teichmuller space of S naturally embeds into the Hitchin component. The limit points in the Thurston compactification of the Teichmuller space are well-understood. Our main goal is to provide non-trivial sufficient conditions on a sequence of Hitchin representations so that the limit of this sequence in the Parreau boundary can be described as an action on a tree. These non-trivial conditions are given in terms of Fock-Goncharov coordinates on moduli spaces of generic tuples of flags.