Algebraic invariants of orbit configuration spaces in genus zero associated to finite groups

Abstract: We consider orbit configuration spaces associated to finite groups acting freely by orientation preserving homeomorphisms on the $2$-sphere minus a finite number of points. Such action is equivalent to a homography action of a finite subgroup $G\subset \mathrm{PGL}(\mathbb{C}^2)$ on the complex projective line $\mathbb{P}^1$ minus a finite set $Z$ stable under $G$. We compute the cohomology ring and the Poincar\'e series of the orbit configuration space $C_n^G(\mathbb{P}^1 \setminus Z)$. This can be seen as a generalization of the work of Arnold for the classical configuration space $C_n(\mathbb{C})$ ($(G,Z)=({1},\infty$)). It follows from the work that $C_n^G(\mathbb{P}^1\setminus Z)$ is formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincar\'e series of $C_n^G(\mathbb{P}^1\setminus Z)$ to the ranks of quotients of successive terms of the lower central series of the fundamental group of $C_n^G(\mathbb{P}^1 \setminus Z)$. The successive quotients correspond to homogenous elements of graded Lie algebras introduced by the author in an earlier work. Such formula is also known for classical configuration spaces of $\mathbb{C}$, where fundamental groups are Artin braid groups and the ranks correspond to dimensions of homogenous elements of the Kohno-Drinfeld Lie algebras.