Peer-reviewedCC BYGroups, Geometry, and Dynamics
On commutator length in free groups
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.
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BibTeX
@article{bartholdi2026commutator,
title = {{On commutator length in free groups}},
author = {Laurent Bartholdi and Sergei O. Ivanov and Danil Fialkovski},
journal = {Groups, Geometry, and Dynamics},
year = {2026},
month = nov,
doi = {10.4171/ggd/747},
url = {https://hal.science/hal-05849733v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - On commutator length in free groups
AU - Laurent Bartholdi
AU - Sergei O. Ivanov
AU - Danil Fialkovski
PY - 2026
DA - 2026/11/03
JO - Groups, Geometry, and Dynamics
DO - 10.4171/ggd/747
UR - https://hal.science/hal-05849733v1
AB - 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.
SN - 1661-7207
SN - 1661-7215
ER -