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

TL;DR

This paper introduces a new class of surface group representations into complex projective linear groups, generalizing pleated surfaces, and provides a holomorphic parametrization extending previous work on pleated surfaces and Hitchin representations.

Contribution

It defines and parametrizes $( ext{lambda},d)$-pleated surfaces arising from generic $ ext{lambda}$-Borel Anosov representations, extending classical and Hitchin case results.

Findings

Holomorphic parametrization of $( ext{lambda},d)$-pleated surfaces.

Extension of Bonahon's work on pleated surfaces.

Connection to Hitchin representations.

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 $\lambda$-Borel Anosov representations, which are representations that are Borel Anosov with respect to a maximal geodesic lamination $\lambda$. For fixed $\lambda$ and $d$, we provide a holomorphic parametrization of the space $\mathcal{R}(\lambda,d)$ of $(\lambda,d)$-pleated surfaces which extends both work of Bonahon for pleated surfaces and Bonahon and Dreyer for Hitchin representations.