TOPOLOGY AND CLASSICAL GEOMETRY IN (2+1)GRAVITY

The structure of the spacetime geometry in (2 + 1) gravity is describe d by means of a foliation in which the space-like surfaces admit a tes sellation made of polygons. The dynamics of the system is determined b y a set of 't Hooft's rules which specify the time evolution of the te ssellation. We illustrate how the non-trivial topology of the universe can be described by means of 't Hooft's formalism. The classical geom etry of a universe with the spatial topology of a torus is considered and the relation between 't Hooft's transitions and modular transforma tions is discussed. The universal covering of spacetime is constructed . The non-trivial topology of an expanding universe gives origin to a redshift effect; we compute the value of the corresponding 'Hubble's c onstant'. Simple examples of tessellations for universes with the spat ial topology of a surface with higher genus are presented.