Enumerating Finite Braid Group Orbits on $SL_2(\C)$-Character Varieties

TL;DR

This paper classifies finite orbits of the braid group action on $SL_2( ext{C})$-character varieties of punctured spheres, combining theoretical analysis with computational methods to identify and construct such orbits.

Contribution

It explicitly classifies finite orbits using middle convolution, linking them to complex reflection groups, and provides computational tools for further exploration.

Findings

Finite orbits are classified via middle convolution and complex reflection groups.

Explicit formulas for constructing finite orbits are provided.

Computational results include exhaustive searches for primitive groups.

Abstract

We analyze finite orbits of the natural braid group action on the character variety of the $n$ times punctured sphere. Building on recent results relating middle convolution and finite complex reflection groups, our work implements Katz's middle convolution to explicitly classify finite orbits in the $SL_2(\C)$-character variety of the punctured sphere. We provide theoretical results on the existence of finite orbits arising from the imprimitive finite complex reflection groups and formulas for constructing such examples when they exist. In the primitive finite complex reflection groups, we perform an exhaustive search and provide computational results. Our contributions also include Magma computer code for middle convolution and for computing the orbit under this action when it is known to be finite.