Presentations of subgroups of the braid group generated by powers of band generators

TL;DR

This paper investigates the structure of subgroups of the braid group generated by powers of band generators, showing they are right-angled Artin groups under certain conditions and providing presentations in other cases.

Contribution

It extends the understanding of subgroup presentations in braid groups, especially for powers of band generators, and characterizes when these subgroups are right-angled Artin groups.

Findings

Subgroups generated by powers of band generators are right-angled Artin groups if all generators have exponent at least 3.

Provides explicit presentations for subgroups with generators of exponent 1 or 2.

Shows these subgroups are not right-angled Artin groups in the case of mixed exponents.

Abstract

According to the Tits conjecture proved by Crisp and Paris, [CP], the subgroups of the braid group generated by proper powers of the Artin elements are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups. The case of subgroups generated by powers of the band generators is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with exponent 1 or 2, which is far from being that of a right-angled Artin group.