On stable commutator length of non-filling curves in surfaces

TL;DR

This paper provides a new proof demonstrating that the stable commutator length of non-filling curves in surface groups is rational, and shows these elements admit extremal surfaces, extending to non-filling 1-chains.

Contribution

It introduces a novel proof of the rationality of scl for non-filling curves and extends the results to non-filling 1-chains in surface groups.

Findings

Stable commutator length (scl) of non-filling curves is rational.

Non-filling curves admit extremal surfaces for scl.

Results extend to non-filling 1-chains.

Abstract

We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also hold more generally for non-filling 1-chains.