Polyhedral Realization of a Thurston Compactification

Let Σ−3 be the connected sum of three real projective planes. We realize the Thurston compactification of the Teichmuller space Teich(Σ−3 ) as a simplex in P(R). First, we define a map L2 from Teich(Σ − 3 ) into P(R ) in terms of some geodesic-length functions. We then introduce the similar triangle flow on Teich(Σ−3 ) to control the ratios between these lengths, and show that L2 is an embedding. Finally, we study the natural extension of L2 to the Thurston boundary using a triangulation of the projective space of measured foliations.