  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.
