Groups are one of the central concepts of modern algebra. The study of the commutator length of elements in different groups is carried out in various areas of mathematics. In particular, information on the commutator length of elements of algebraic groups is used in algebraic K-theory. Researches in this area were conducted by such scientists as C.Edmunds, R. Golstein, E. Turner, M. Culler, L. Comerford, D. Calegari, V. Bardakov and others.
It is well known that any group can be represented as a factor in a group of a free group, and under a homomorphism the commutator length does not increase. Therefore, the commutator length of elements of a free group is of particular interest.
We propose a fast algorithm for calculating the switch length of an element from the commutant of a free group. This algorithm relies on the already existing Bardakov algorithm, which, unlike the one we have proposed, does not explicitly represent an element as a product of commutators. In addition, our algorithm is faster.
Also, on the basis of my algorithm, a program can be written that makes it possible to read the switching length of an element and get an explicit representation of the product of switches faster and more efficiently than any of the existing ones.
Source: https://habr.com/ru/post/367145/