How can we tell if a group element is a k-fold nested commutator? I suggest to seek computable invariants of words in groups that detect k-fold commutators.

We introduce the novel theory of letter-braiding invariants: these are elementarily defined functions on words, inspired by the homotopy theory of loop-spaces and carrying deep geometric content. They give a universal finite-type invariant for arbitrary groups, extending the influential Magnus expansion of free groups that already had countless applications in low dimensional topology.

As a consequence we get new combinatorial formulas for braid and link invariants, and a way to linearlize automorphisms of general groups that specializes to the Johnson homomorphism of mapping class groups. This theory shows potential to advance a wide range of areas, from finite group theory to intersection theory in manifolds and p-adic Hodge theory of fundamental groups.