Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given.
"Applications of combinatorial groups to Hopf invariant and the exponent problem." Algebr. Geom. Topol. 6 (5) 2229 - 2255, 2006. https://doi.org/10.2140/agt.2006.6.2229