The torus universe in the polygon approach to (2 + 1)-dimensional gravity

In this paper we describe the matter-free toroidal spacetime in 't Hooft's polygon approach to (2 + 1)-dimensional gravity. First we show that the constraint algebra of the polygons closes (this is a general result, not necessarily derived for a torus). Next we construct a one-polygon torus and find (in contrast to earlier results in the literature) that this slicing of spacetime is not compatible with all the solutions that emerge in the continuum formulation. Finally, we remedy this situation by adding one more polygon.