Hitchin grafting representations II: Dynamics

TL;DR

This paper explores the dynamics within the Hitchin component of surface group representations, revealing that certain deformation paths have finite length in the pressure metric despite infinite diameter in the Weil--Petersson metric.

Contribution

It demonstrates the existence of quasi-convex subsets with infinite Weil--Petersson diameter but finite pressure metric length in the Hitchin component for genus at least 3.

Findings

Quasi-convex subsets with infinite Weil--Petersson diameter

Finite length of bending deformation paths in pressure metric

Controlled bounded length of biinfinite bending paths

Abstract

The Hitchin component of the character variety of representations of a surface group $\pi_1(S)$ into $\mathrm{PSL}_d(\mathbb{R})$ for some $d \geq 3$ can be equipped with a pressure metric whose restriction to the Fuchsian locus equals the Weil--Petersson metric up to a constant factor. We show that if the genus of $S$ is at least $3$, then the Fuchsian locus contains quasi-convex subsets of infinite diameter for the Weil--Petersson metric whose diameter for the path metric of the pressure metric is finite. This is established through showing that biinfinite paths of bending deformations have controlled bounded length.