The center of pure complex braid groups

TL;DR

This paper proves that the center of pure complex braid groups associated with irreducible finite complex reflection groups is cyclic, extending known results and establishing that finite index subgroups also have cyclic centers.

Contribution

It confirms Broué, Malle, and Rouquier's conjecture for all exceptional types and shows that all finite index subgroups of these braid groups have cyclic centers.

Findings

Confirmed cyclicity of the center for all exceptional types

Extended the result to finite index subgroups

Strengthened the understanding of the structure of pure complex braid groups

Abstract

Brou\'e, Malle and Rouquier conjectured in that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for the full braid group due to Bessis, and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center.