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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Faces of the scl norm ball

Danny Calegari

Geometry & Topology 13 (2009) 1313–1336

Let F = π1(S) where S is a compact, connected, oriented surface with χ(S) < 0 and nonempty boundary.

(1)  The projective class of the chain S B1H(F) intersects the interior of a codimension one face πS of the unit ball in the stable commutator length norm on B1H(F).

(2)  The unique homogeneous quasimorphism on F dual to πS (up to scale and elements of H1(F)) is the rotation quasimorphism associated to the action of π1(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.

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
Received: 22 July 2008
Revised: 19 January 2009
Accepted: 17 January 2009
Published: 13 February 2009
Proposed: Benson Farb
Seconded: Dmitri Burago, Leonid Polterovich
Danny Calegari
Department of Mathematics
Pasadena, CA 91125