Automorphisms of pure braid Groups

Abstract

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.

17 pp

References in corpus (1)

• Automorphism groups of some pure braid groups

Cited by in corpus (3)

• Some remarks on twin groups
• The R${}_{\infty }$ property for pure Artin braid groups
• Lower central series, surface braid groups, surjections and permutations