Hitchin grafting representations II: Dynamics

Abstract

The Hitchin component of the character variety of representations of a surface group $\ddot{I}{€}_1(S)$ into ${\mathrm{PSL}}_d(\mathbb{R})$ for some $d\ge 3$ 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.

This paper was split into two papers, and this is the second part. The first part is the paper called "Geometry and dynamics of Hitchin grafting representations I: Geometry". The proof of the estimates on derivatives of lengths (now section 5) was changed (the previous proof had a flaw). Other small corrections and changes were made. 74 pages, 2 figures, comments welcome!