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

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.