New systems of log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties of compact Riemann surfaces

TL;DR

The paper introduces new log-canonical coordinate systems on $SL(2, \, \mathbb{C})$ character varieties of compact Riemann surfaces, generalizing Fenchel-Nielsen coordinates via loop-based labellings.

Contribution

It constructs novel coordinate systems combining shear and length/twist types, linked to trinion decompositions and extending Fenchel-Nielsen coordinates.

Findings

Coordinates are labeled by families of non-intersecting loops.

Coordinates unify shear-type and length/twist-type parameters.

In the maximal case, they relate closely to Fenchel-Nielsen coordinates.

Abstract

We construct new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\leq m\leq 3g-3$ non-intersecting simple loops on the Riemann surface and are obtained by combining the complexified shear-type with length/twist-type coordinates. In the case $m=3g-3$ the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones.