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

Daciberg Lima Gonçalves (1) , John Guaschi (2)

1 IME - Instituto de Matemática e Estatística
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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

Mots clés: surface braid groups; virtually cyclic groups; projective plane; torsion elements

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

Daciberg Lima Gonçalves, John Guaschi. Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane. Journal of Group Theory, 2010, 13 (2), pp.277-294. ⟨10.1515/JGT.2009.040⟩. ⟨hal-00184740⟩