Fenchel-Nielsen coordinates for SL(3,C) representations

TL;DR

This paper introduces Fenchel-Nielsen coordinates for surface group representations into SL(3,C), linking them to classical and generalized coordinates, thereby expanding the geometric framework for higher rank Lie groups.

Contribution

It defines Fenchel-Nielsen coordinates for SL(3,C) representations and relates them to existing classical and generalized Fenchel-Nielsen coordinates.

Findings

Established a new coordinate system for SL(3,C) representations.

Connected the new coordinates to classical Fenchel-Nielsen coordinates.

Extended the geometric understanding of surface group representations.

Abstract

We define Fenchel-Nielsen coordinates for representations of surface groups to ${\rm SL}(3,{\mathbb C})$. We also show how these coordinates relate to the classical Fenchel-Nielsen coordinates and to their generalisations by Kourouniotis, Tan, Goldman, Zhang and Parker-Platis.