Hitchin grafting representations I: Geometry

TL;DR

This paper provides a geometric interpretation of Fock--Goncharov positivity and demonstrates how bending deformations of Fuchsian representations stabilize a specific Finsler quasi-convex disk in a symmetric space.

Contribution

It introduces a new geometric perspective on Fock--Goncharov positivity and analyzes the stabilization of Finsler disks by bending deformations of Fuchsian representations.

Findings

Fock--Goncharov positivity has a geometric interpretation.

Bending deformations stabilize a Finsler quasi-convex disk.

The stabilized disk resides in the symmetric space PSL_d(R)/PSO(d).

Abstract

We give a geometric interpretation of Fock--Goncharov positivity and show that bending deformations of Fuchsian representations stabilize a uniform Finsler quasi-convex disk in the symmetric space $\mathrm{PSL}_d(\mathbb R)/\mathrm{PSO}(d)$.