Labeled binary planar trees and quasi-Lie algebras

Abstract

We study the natural map $\eta$ between a group of binary planar trees whose leaves are labeled by elements of a free abelian group $H$ and a certain group $\mathsf{D}\left(H\right)$ derived from the free Lie algebra over $H$. Both of these groups arise in several different topological contexts. $\eta$ is known to be an isomorphism over $\mathbb{Q}$, but not over $\mathbb{Z}$. We determine its cokernel and attack the conjecture that it is injective.