Work

On commutator length in free groups

Year: 2023

Type: article

Abstract: Let F be a free group. We present for arbitrary g\in\mathbb{N} a LOGSPACE (and thus polynomial time) algorithm that determines whether a given w\in F is a product of at most g commutators; and more generally, an algorithm that determines, given w\in F , the minimal g such that w may be written as a product of g commutators (and returns \infty if no such g exists). This algorithm also returns words x_1,y_1,\dots,x_g,y_g such that w=[x_1,y_1]\dots[x_g,y_g] . These algorithms are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a conjecture by Bardakov.

Source: Groups Geometry and Dynamics

DOI: 10.4171/ggd/747

Authors: Laurent Bartholdi, Sergei O. Ivanov, Danil Fialkovski

Institutions: Saarland University, St Petersburg University, Beijing Institute of Mathematical Sciences and Applications

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Ministry of Science and Higher Education of the Russian Federation

Awards: G2337782072, G6444257319