Faces of the scl norm ball

Abstract

Let $F={\pi }_1\left(S\right)$ where $S$ is a compact, connected, oriented surface with $\chi \left(S\right)<0$ and nonempty boundary.

(1) The projective class of the chain $\partial S\in B_1^H\left(F\right)$ intersects the interior of a codimension one face ${\pi }_S$ of the unit ball in the stable commutator length norm on $B_1^H\left(F\right)$.

(2) The unique homogeneous quasimorphism on $F$ dual to ${\pi }_S$ (up to scale and elements of $H^1\left(F\right)$) is the rotation quasimorphism associated to the action of ${\pi }_1\left(S\right)$ 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 $\partial S$. This is true even if $S$ has no boundary.