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

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 $\Z_{4} \ast_{\Z_{2}} \Z_{4}$.