Abstract

The first main objective of this thesis is to study the orbit structure of the (2 + 1)-Poincare group R2,1 [Special characters omitted.] 2,1 {604} SO (2,1) by obtaining an explicit expression for the coadjoint action. From there, we compute and classify the coadjoint orbits. We obtain a degenerate orbit, the upper and lower sheet of the two-sheet hyperboloid, the upper and lower cone and the one-sheet hyperboloid. They appear as two-dimensional coadjoint orbits and, with their cotangent planes, as four-dimensional coadjoint orbits. We also confirm a link between the four-dimensional coadjoint orbits and the orbits of the action of SO (2, 1) on the dual of [Special characters omitted.] 2,1 . The second main objective of this thesis is to use the information obtained about the structure to induce a representation and build the coherent states on two of the coadjoint orbits. We obtain coherent states on the hyperboloid for the principal section. The Galilean and the affine sections only allow us to get frames. On the cone, we obtain a family of coherent states for a generalized principal section and a frame for the basic section