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

TL;DR

This paper classifies all virtually cyclic subgroups of pure braid groups on the real projective plane, identifying their finite and infinite types and their specific structures for various n.

Contribution

It provides a complete classification of virtually cyclic subgroups of pure braid groups on the projective plane, detailing their finite and infinite cases and maximal finite subgroups.

Findings

Maximal finite subgroups are quaternion group of order 8 for n=3

Maximal finite subgroups are cyclic of order 4 for n≥4

Infinite virtually cyclic subgroups include Z, Z2×Z, and a specific amalgamated product

Abstract

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to $\Z_{4}$ if $n\geq 4$. Further, for all $n\geq 3$, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of $P_{n}(RP^2)$: $\Z$, $\Z_{2} \times \Z$ and the amalgamated product $\Z_{4} \ast_{\Z_{2}} \Z_{4}$.