On the Complexity of Sails

Abstract

This paper analyses stable commutator length in groups Z^r * Z^s. We bound scl from above in terms of the reduced wordlength (sharply in the limit) and from below in terms of the answer to an associated subset-sum type problem. Combining both estimates, we prove that, as n tends to infinity, words of reduced length n generically have scl arbitrarily close to n/4 - 1. We then show that, unless P=NP, there is no polynomial time algorithm to compute scl of efficiently encoded words in F2. All these results are obtained by exploiting the fundamental connection between scl and the geometry of certain rational polyhedra. Their extremal rays have been classified concisely and completely. However, we prove that a similar classification for extremal points is impossible in a very strong sense.