Faces of the scl norm ball

Abstract

Let F = π_1(S) where S is a compact, connected, oriented surface with χ(S) < 0 and nonempty boundary. (1) The projective class of the chain ∂S ∈ B^H_1(F) intersects the interior of a codimension one face π_S of the unit ball in the stable commutator length norm on B^H_1(F). (2) The unique homogeneous quasimorphism on F dual to π_S (up to scale and elements of H^1(F)) is the rotation quasimorphism associated to the action of π_1(S) on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on S. These facts follow from the fact that every homologically trivial 1–chain C in S rationally cobounds an immersed surface with a sufficiently large multiple of ∂S. This is true even if S has no boundary.