Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

Abstract

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_n(R{P}^2)$ of the projective plane. The maximal finite subgroups of $P_n(R{P}^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to ${\mathbb{Z}}_4$ if $n\ge 4$. Further, for all $n\ge 3$, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of $P_n(R{P}^2)$: $\mathbb{Z}$, ${\mathbb{Z}}_2\times \mathbb{Z}$ and the amalgamated product ${\mathbb{Z}}_4{\ast }_{{\mathbb{Z}}_2}{\mathbb{Z}}_4$.

15 pages

References in corpus (4)

• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• Configuration spaces of C and CP^1: some analytic properties
• Conjugacy classes of finite subgroups of certain mapping class groups
• The quaternion group as a subgroup of the sphere braid groups

Cited by in corpus (3)

• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• The classification of the virtually cyclic subgroups of the sphere braid groups
• Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces