Faces of the scl norm ball

摘要

Let F velence (pi)_(1)(S) where S is a compact, connected, oriented surface with chi(S) < 0 and nonempty boundary. (1) The projective class of the chain (partial deriv)S E B_(1)~(H)(F) intersects the interior of a codimension one face (pi)_(S) of the unit ball in the stable commutator length norm on B_(1)~(H)(F). (2) The unique homogeneous quasimorphism on F dual to (pi)_(S) (up to scale and elements of H~(1)(F)) is the rotation quasimorphism associated to the action of (pi)_(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 (partial deriv)S. This is true even if S has no boundary.