$d$-pleated surfaces and their shear-bend coordinates

Abstract

In this article, we single out representations of surface groups into ${\mathsf{PSL}}_d(\mathbb{C})$ which generalize the well-studied family of pleated surfaces into ${\mathsf{PSL}}_2(\mathbb{C})$. Our representations arise as sufficiently generic $\hat{I}»$-Borel Anosov representations, which are representations that are Borel Anosov with respect to a maximal geodesic lamination $\hat{I}»$. For fixed $\hat{I}»$ and $d$, we provide a holomorphic parametrization of the space $\mathcal{R}(\hat{I}»,d)$ of $(\hat{I}»,d)$-pleated surfaces which extends both work of Bonahon for pleated surfaces and Bonahon and Dreyer for Hitchin representations.

134 pages, 13 figures. Exposition reorganized following peer-review. Added explicit examples of classical pleated surfaces with reducible holonomy, see Section 12.1