Stable curves and screens on fatgraphs

The mapping class group invariant ideal cell decomposition of the Teichmueller space of a punctured surface times an open simplex has been used in a number of computations. This paper answers a question about the asymptotics of this decomposition, namely, in a given cell of the decomposition, which curves can be short? Screens are a new combinatorial structure which provide an answer to this question. The heart of the calculation here involves Ptolemy transformations and the triangle inequalities on lambda lengths.