Faces of the scl norm ball

Calegari, Danny

doi:10.2140/gt.2009.13.1313

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Faces of the scl norm ball

Danny Calegari

Geometry & Topology 13 (2009) 1313–1336

DOI: 10.2140/gt.2009.13.1313

Abstract

Let $F = π_{1} (S)$ where $S$ is a compact, connected, oriented surface with $χ (S) < 0$ and nonempty boundary.

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

(2) The unique homogeneous quasimorphism on $F$ dual to $π_{S}$ (up to scale and elements of $H^{1} (F)$ ) is the rotation quasimorphism associated to the action of $π_{1} (S)$ 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 $\partial S$ . 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

References

Publication

Received: 22 July 2008

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

Volume 13, issue 3 (2009)