  There has been strong interest in the possibility that in the quantum-gravity
realm momentum space might be curved, mainly focusing, especially for what
concerns phenomenological implications, on the case of a de Sitter momentum
space. We here take as starting point the known fact that quantum gravity
coupled to matter in $2+1$ spacetime dimensions gives rise to an effective
picture characterized by a momentum space with anti-de Sitter geometry, and we
point out some key properties of $2+1$-dimensional anti-de Sitter momentum
space. We observe that it is impossible to implement all of these properties in
theories with a $3+1$-dimensional anti-de Sitter momentum space, and we then
investigate, with the aim of providing guidance to the relevant phenomenology
focusing on possible modified laws of conservation of momenta, the implications
of giving up, in the $3+1$-dimensional case, some of the properties of the
$2+1$-dimensional case.
