THE POLYGON REPRESENTATION OF FLAT 3-DIMENSIONAL SPACETIMES

An important set of solutions of 2 + 1 dimensional gravity consists of spacetimes which contain an open neighborhood which admits a slicing SIGMA(g,N,b) x (0, 1) in a family of genus g Riemann surfaces with N punctures and b asymptotic regions. We define a set of polygons P, an equivalence relation R, and a bijective map from P/R to this set of spacetimes. The construction of a spacetime from the corresponding polygon leads to a natural extension beyond singular surfaces. We discuss the existence of a global spacelike foliation, and generalize to de Sitter manifolds, or 2 + 1-dimensional gravity with a cosmological constant.