On homomorphisms of $\pi_{1}(\mathbb P^1-\mathcal R)$ into compact semisimple groups

TL;DR

This paper establishes verifiable criteria for the existence of irreducible homomorphisms from the fundamental group of a punctured projective line into compact semisimple groups, linking the problem to stable torsors on the complex projective line.

Contribution

It provides new verifiable conditions for irreducible homomorphisms into compact semisimple groups, connecting group homomorphisms with stable torsors via Bruhat-Tits group schemes.

Findings

Criteria for irreducible homomorphisms are established.

Reduction of the problem to stable torsors on 1_{\u00A9}C.

Connection with Bruhat-Tits group schemes.

Abstract

The aim of this paper is to give verifiable criteria for the existence of {\em irreducible} homomorphisms of $\pi_{1}(\mathbb P^1 - \mathcal R)$ into compact semisimple groups, for a finite subset $\mathcal R$ such that the conjugacy classes of the images of lassos around the marked points are fixed. By a theorem in \cite{bs}, this question reduces into one of giving verifiable criteria for the existence of stable $\mathcal G$-torsors on $\mathbb{P}^1_{\mathbb{C}}$, where $\mathcal{G} \to \mathbb{P}^1_\mathbb{C}$ is a a Bruhat-Tits group scheme.