Thurston introduced pleated surfaces as a powerful tool to study hyperbolic 3-manifolds.
An abstract pleated surface is a representation of the fundamental group of a hyperbolic surface into the Lie group PSL(2,C) of orientation
preserving isometries of hyperbolic 3-space together with an equivariant map from the hyperbolic plane into hyperbolic 3-space which satisfies additional properties.

In this talk, we introduce a notion of d-pleated surface for representations into PSL(d,C) which is motivated by the theory of Anosov
representations. In addition, we give a holomorphic parametrization of the space of d-pleated surfaces via cocyclic pairs, thus generalizing a result of Bonahon.

This talk is based on joint work with Sara Maloni, Filippo Mazzoli and Tengren Zhang.