1Universität des Saarlandes, Saarbrücken, Germany 2St. Petersburg State University, Russia; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 3St. Petersburg State University, Russia
Groups, geometry, and dynamics, Tome 18 (2024) no. 1, pp. 191-202
Let F be a free group. We present for arbitrary g∈N a LOGSPACE (and thus polynomial time) algorithm that determines whether a given w∈F is a product of at most g commutators; and more generally, an algorithm that determines, given w∈F, the minimal g such that w may be written as a product of g commutators (and returns ∞ if no such g exists). This algorithm also returns words x1,y1,...,xg,yg such that w=[x1,y1]...[xg,yg]. 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.
Classification :
20-XX
Mots-clés :
Commutator length, equations in free groups
Affiliations des auteurs :
Laurent Bartholdi
1
;
Sergei O. Ivanov
2
;
Danil Fialkovski
3
1
Universität des Saarlandes, Saarbrücken, Germany
2
St. Petersburg State University, Russia; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing
3
St. Petersburg State University, Russia
Laurent Bartholdi; Sergei O. Ivanov; Danil Fialkovski. On commutator length in free groups. Groups, geometry, and dynamics, Tome 18 (2024) no. 1, pp. 191-202. doi: 10.4171/ggd/747
@article{10_4171_ggd_747,
author = {Laurent Bartholdi and Sergei O. Ivanov and Danil Fialkovski},
title = {On commutator length in free groups},
journal = {Groups, geometry, and dynamics},
pages = {191--202},
year = {2024},
volume = {18},
number = {1},
doi = {10.4171/ggd/747},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/747/}
}
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