Scl, sails and surgery

Given a group $G$ and an element $g \in [G,G]$, the commutator length of $g$, denoted $\cl(g)$, is the smallest number of commutators in $G$ whose product is $g$, and the stable commutator length of $g$ is the limit $scl(g):=\lim_{n to \infty} \cl(g^n)/n$. Commutator length in a group extends in a natural way to a pseudo-norm on the real vector space of $1$-boundaries (in group homology), and should be thought of as a kind of relative Gromov-Thurston norm. We show that the problem of computing stable commutator length in free products of abelian groups reduces to a (finite dimensional) integer programming problem. Moreover, certain families of elements in such groups (i.e. those obtained by surgery on some element in a bigger group) give rise to families of integer programming problems that are related in explicit ways. In particular one can use this to establish the existence of limit points in the range of scl in such groups, and produce elements whose stable commutator length is congruent to any rational number modulo the integers. This technology relates stable commutator length to the theory of multi-dimensional continued fractions, and Klein polyhedra, and suggests an interesting conjectural picture of scl in free groups.