#### Volume 13, issue 3 (2009)

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Faces of the scl norm ball

### Danny Calegari

Geometry & Topology 13 (2009) 1313–1336
##### 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.

##### Keywords
immersion, surface, free group, bounded cohomology, scl, polyhedral norm, rigidity, hyperbolic structure, rotation number
##### Mathematical Subject Classification 2000
Primary: 20F65, 20J05
Secondary: 20F67, 20F12, 55N35, 57M07
##### Publication
Revised: 19 January 2009
Accepted: 17 January 2009
Published: 13 February 2009
Proposed: Benson Farb
Seconded: Dmitri Burago, Leonid Polterovich
##### Authors
 Danny Calegari Department of Mathematics Caltech Pasadena, CA 91125 USA http://www.its.caltech.edu/~dannyc