Finite braid group orbits on $SL_2$-character varieties

TL;DR

This paper classifies certain rank 2 local systems on punctured spheres with infinite order monodromy, showing they are either pullback types or derived from finite complex reflection groups, enriching the understanding of character varieties.

Contribution

It provides a complete classification of Zariski-dense representations with finite orbits under the mapping class group, especially those with infinite order local monodromy, via pullback and middle convolution methods.

Findings

Classified all conjugacy classes of such representations.

Identified all rank 2 local systems of geometric origin with specified monodromy.

Connected the representations to finite complex reflection groups.

Abstract

Let X be a 2-sphere with n punctures. We classify all conjugacy classes of Zariski-dense representations $$\rho: \pi_1(X)\to SL_2(\mathbb{C})$$ with finite orbit under the mapping class group of X, such that the local monodromy at one or more punctures has infinite order. We show that all such representations are "of pullback type" or arise via middle convolution from finite complex reflection groups. In particular, we classify all rank 2 local systems of geometric origin on the projective line with n generic punctures, and with local monodromy of infinite order about at least one puncture.