Space-time defects and group momentum space

We study massive and massless conical defects in Minkowski and de Sitter spaces in various spacetime 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 spacetime 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 n+1-dimensional Minkowski space, then its space of holonomies is a maximal abelian subgroup of the AN(n-1) group, which corresponds to the well known momentum space associated with the n-dimensional $kappa$-Minkowski noncommutative spacetime and $kappa$-deformed Poincar'e algebra. We also conjecture that massless defects in n-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 spacetime defects.