Topology and classical geometry in (2+1) gravity

The structure of the spacetime geometry in (2 + 1) gravity is described by means of a foliation in which the space-like surfaces admit a tessellation made of polygons. The dynamics of the system is determined by a set of 't Hooft's rules which specify the time evolution of the tessellation. We illustrate how the non-trivial topology of the universe can be described by means of 't Hooft's formalism. The classical geometry of a universe with the spatial topology of a torus is considered and the relation between 't Hooft's transitions and modular transformations 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 constant'. Simple examples of tessellations for universes with the spatial topology of a surface with higher genus are presented. © 1996 IOP Publishing Ltd.