Orbit configuration spaces associated to discrete subgroups of PSL(2,R)

TL;DR

This paper investigates Lie algebras arising from orbit configuration spaces linked to discrete subgroups of PSL(2,R), revealing their isomorphisms with algebras from higher homotopy groups and chord diagrams, and exploring related Poisson structures.

Contribution

It establishes isomorphisms between Lie algebras from orbit configuration spaces and those from higher homotopy groups and chord diagrams, extending understanding of their algebraic structures.

Findings

Lie algebra from fundamental group matches that from higher homotopy groups

Identifies isomorphisms up to regrading

Analyzes a graded Poisson algebra related to braid relations

Abstract

The purpose of this article is to analyze several Lie algebras associated to "orbit configuration spaces" obtained from a group G acting freely, and properly discontinuously on the upper 1/2-plane H^2. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to (1) the Lie algebra obtained from the higher homotopy groups of "higher dimensional arrangements" modulo torsion, as well as (2)the Lie obtained from horizontal chord diagrams for surfaces. The resulting Lie algebras are similar to those studied in [13, 14, 15, 2, 7, 8, 6]. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by G is also addressed.