The Pauli - Lubanski scalar in the polygon approach to (2+1)-dimensional gravity

In this paper we derive an expression for the conserved Pauli - Lubanski scalar in 't Hooft's polygon approach to (2 + 1)-dimensional gravity coupled to point particles. We find that it is represented by an extra spatial shift in addition to the usual identification rule (being a rotation over the cut). For two particles this invariant is expressed in terms of 't Hooft's phase-space variables and we check its classical limit.