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(𝑃𝑛) is generated by the subgroup Aut𝑐(𝑃𝑛) of central automorphisms of 𝑃𝑛, the subgroup Aut(𝐵𝑛) of restrictions of automorphisms of 𝐵𝑛 on 𝑃𝑛 and one extra automorphism 𝑤𝑛. 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 𝑃𝑛 can be extended to an automorphism of 𝐵𝑛.