Letter braiding - using algebraic topology to measure words in groups
How can we tell if a group element can be written as a k-fold nested commutator? One approach is to find computable invariants of words in groups that vanish on all (k-1)-fold commutators but not on k-fold ones. We introduce the theory of letter-braiding invariants - these are "polynomial" functions on words, inspired by the homotopy theory of loop-spaces and Koszul duality, and carrying deep geometric content. They extend the influential Magnus expansion of free groups, which already had countless applications in low dimensional topology, into a functor defined on arbitrary groups. As a consequence we get new combinatorial formulas for braid and link invariants, and a way to linearize automorphisms of general groups which specializes to the Johnson homomorphism of mapping class groups.
תאריך עדכון אחרון : 10/06/2025