Sur certains espaces de configurations associ\'es aux sous-groupes finis de $\mathrm{PSL}_2(\mathbb{C}) $

TL;DR

This paper investigates orbit configuration spaces arising from finite homography group actions on the projective line, constructing flat connections and establishing isomorphisms between associated Lie algebras and fundamental groups.

Contribution

It introduces a flat connection on orbit spaces and proves isomorphisms between their Lie algebras and fundamental groups, linking geometric and algebraic structures.

Findings

Constructed a flat connection on orbit configuration spaces.

Established isomorphisms between Lie algebras and fundamental groups.

Connected monodromy representations with algebraic structures.

Abstract

We study orbit configuration spaces $\mathrm{Cf}_G(n,\mathbb{P}^1_*)$ obtained from the action of a finite homography group $G$ on $\mathbb{P}^1$. We construct a flat connection on the orbit space with values in a Lie algebra $\hat{\mathfrak{p}}_n(G) $. We establish an isomorphism of filtered Lie algebras between $\hat{\mathfrak{p}}_n(G)$, the Malcev Lie algebra of the fundamental group of $\mathrm{Cf}_G(n,\mathbb{P}^1_*)$ and the degree completion of the associated graded to the latter Lie algebra. These isomorphisms are obtained using the monodromy representation of the connection and the study of the fundamental group.