Let where is a compact, connected, oriented surface with and nonempty boundary.
(1) The projective class of the chain intersects the interior of a codimension one face of the unit ball in the stable commutator length norm on .
(2) The unique homogeneous quasimorphism on dual to (up to scale and elements of ) is the rotation quasimorphism associated to the action of on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on .
These facts follow from the fact that every homologically trivial –chain in rationally cobounds an immersed surface with a sufficiently large multiple of . This is true even if has no boundary.
"Faces of the scl norm ball." Geom. Topol. 13 (3) 1313 - 1336, 2009. https://doi.org/10.2140/gt.2009.13.1313