We study massive and massless conical defects in Minkowski and de Sitter spaces in various space-time dimensions. The energy momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its space-time metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects, respectively. In particular, if we fix the direction of propagation of a massless defect in -dimensional Minkowski space, then its space of holonomies is a maximal Abelian subgroup of the AN group, which corresponds to the well known momentum space associated with the -dimensional -Minkowski noncommutative space-time and -deformed Poincaré algebra. We also conjecture that massless defects in -dimensional de Sitter space can be analogously characterized by holonomies belonging to the same subgroup. This shows how group-valued momenta related to four-dimensional deformations of relativistic symmetries can arise in the description of motion of space-time defects.