On the Complexity of Sails

TL;DR

This paper investigates the stable commutator length in certain groups, providing bounds, asymptotic behavior, and complexity results, by linking the problem to the geometry of rational polyhedra and subset-sum problems.

Contribution

It establishes bounds and asymptotic behavior of scl in free abelian groups, and proves the computational hardness of calculating scl in free groups, using geometric and combinatorial methods.

Findings

scl is asymptotically close to n/4 - 1 for generic words of length n

computing scl in free groups is NP-hard unless P=NP

classification of extremal rays of associated polyhedra is complete, but extremal points classification is impossible

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.