Deformed Carroll particle from 2+12+1 gravity

We consider a point particle coupled to 2+12+1 gravity, with de Sitter gauge group SO(3,1)SO(3,1). We observe that there are two contraction limits of the gauge group: one resulting in the Poincaré group, and the second with the gauge group having the form AN(2)⋉an(2)⁎AN(2)⋉an(2)⁎. The former case was thoroughly discussed in the literature, while the latter leads to the deformed particle action with de Sitter momentum space, like in the case of κ  -Poincaré particle. However, the construction forces the mass shell constraint to have the form p02=m2, so that the effective particle action describes the deformed Carroll particle.