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

TL;DR

The paper provides a comprehensive, computation-assisted classification of finite braid-group orbits on SL_2(C) character varieties of punctured spheres, building on Katz’s middle convolution and the theory of finite complex reflection groups. By systematically searching imprimitive and primitive complex reflection groups for “nice” tuples and applying middle convolution, the authors produce explicit rank-2 MC-finite local systems and map their braid-group orbits, including extensive Magma-based implementations. They enumerate which groups yield such tuples (and which do not), present explicit exemplars, and relate many findings to Lisovyy–Tykhyy and Tykhyy classifications of Painlevé VI and related systems, including instances with infinite monodromy. The work thus strengthens the bridge between finite reflection groups, middle convolution, and nonlinear differential-equationsolving frameworks, while providing practical computational tools to generate and verify new finite-orbit examples. It also clarifies the landscape of viable cases across imprimitive and primitive groups and highlights where middle convolution can or cannot produce rank-2, MCG-finite local systems on punctured spheres.

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.

Theorems & Definitions (89)

• Definition 1.1.1
• Definition 1.1.2
• Remark 1.1.3
• Theorem 1.1.4
• Theorem 1.1.5
• Theorem 1.1.6
• Theorem 1.1.7
• Remark 1.1.8
• Theorem 1.1.9
• Theorem 1.1.10