Grupos de tranças virtuais, grupos de tranças planas virtuais e grupos cristalográficos.

Abstract

Let n ≥ 2. Let V Bn (resp. V Pn) be the virtual braid group (resp. the pure virtual braid group), and let V Tn (resp. P V Tn) be the virtual twin group (resp. the pure virtual twin group). Let Π be one of the following quotients: V Bn/Γ2(V Pn) or V Tn/Γ2(P V Tn) where Γ2(H) is the commutator subgroup of H. In this thesis, we show that Π is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in Π. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in Π. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) mo dulo the respective commutator subgroup in each case. Even more, we show that the groups Bn(M)/Γk(Pn(M)), where M is the finitely punctured sphere, V Bn/Γ3(V Pn), V Tn/Γk(P V Tn) and UV Bn/Γk(UV Pn) are almost-crystallographic group.