Forum Problem Lists Definitions FAQ

14.102 (1999)

Partially Solved

(V. Lin). Let $B_n$ be the braid group on $n$ strings, and let $n > 4$.
$\qquad$ a) Does $B_n$ have any non-trivial non-injective endomorphisms with non-cyclic images?
$\qquad$ b) Is it true that every non-trivial endomorphism of the derived subgroup $[B_n, B_n]$ is an automorphism?
$\qquad$ c) Does $B_n$ have proper non-abelian torsion-free factor-groups?

Contributor: V. Shpilrain

Progress

a) Editors' comment: No, for $n \geqslant 5$ any non-injective homomorphism $B_n \to B_n$ has cyclic image (F. Castel, Geometric representations of the braid groups, (Astérisque, 378), Paris, 2016, for $n \geqslant 6$, and L. Chen, K. Kordek, D. Margalit, Preprint, 2019, https://arxiv.org/abs/1910.00712 for $n = 5$).

b) Yes, it is true for $n \geqslant 5$ (S. Orevkov, Ann. Fac. Sci. Toulouse. Math., 33, no. 1 (2024), 105–121, extending the result for $n \geqslant 7$ in K. Kordek, D. Margalit, Bull. London Math. Soc., 54, no. 1 (2022), 95–111).

c) Comment of 2001: S. P. Humphries, (Int. J. Algebra Comput., 11, no. 3 (2001), 363–373) has constructed a representation of $B_n$ which is shown to provide torsion-free non-abelian factor groups of $B_n$ as well as of $[B_n, B_n]$ for $n < 7$. It is likely that the same representation should work for other values of $n$ as well.
c) Yes, it does; moreover, every braid group $B_n$ is residually torsion-free nilpotent-by-finite (P. Linnell, T. Schick, J. Amer. Math. Soc., 20, no. 4 (2007), 1003–1051).

Comments

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic. If you want to submit a proof (or a partial proof), please use the dedicated proof submission form rather than posting it in the comments.
Order by newest first or oldest first.

No comments yet. Be the first to comment.

Proof claims

Proof claims are the responsibility of the submitter. Appearance here does not mean the claim has been checked for mathematical correctness. Moderators only screen for spam, abuse, and obviously low-effort submissions.

No proof claims yet.