## Geometry & Topology

### Faces of the scl norm ball

Danny Calegari

#### Abstract

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.

#### Article information

Source
Geom. Topol., Volume 13, Number 3 (2009), 1313-1336.

Dates
Revised: 19 January 2009
Accepted: 17 January 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800248

Digital Object Identifier
doi:10.2140/gt.2009.13.1313

Mathematical Reviews number (MathSciNet)
MR2496047

Zentralblatt MATH identifier
1228.20032

#### Citation

Calegari, Danny. Faces of the scl norm ball. Geom. Topol. 13 (2009), no. 3, 1313--1336. doi:10.2140/gt.2009.13.1313. https://projecteuclid.org/euclid.gt/1513800248

#### References

• C Bavard, Longueur stable des commutateurs, Enseign. Math. $(2)$ 37 (1991) 109–150
• S Blank, Extending immersions of the circle, PhD thesis, Brandeis University (1967)
• M Burger, A Iozzi, A Wienhard, Surface group representations with maximal Toledo invariant, C. R. Math. Acad. Sci. Paris 336 (2003) 387–390
• D Calegari, scallop, Computer program Available at \setbox0\makeatletter\@url http://www.its.caltech.edu/~dannyc {\unhbox0
• D Calegari, scl, Monograph Available at \setbox0\makeatletter\@url http://www.its.caltech.edu/~dannyc/scl/toc.html {\unhbox0
• D Calegari, Stable commutator length is rational in free groups
• D Calegari, Universal circles for quasigeodesic flows, Geom. Topol. 10 (2006) 2271–2298
• D Calegari, Surface subgroups from homology, Geom. Topol. 12 (2008) 1995–2007
• G B Dantzig, Linear programming and extensions, Princeton Univ. Press (1963)
• G K Francis, Spherical curves that bound immersed discs, Proc. Amer. Math. Soc. 41 (1973) 87–93
• W Goldman, Discontinuous groups and the Euler class, PhD thesis, UC Berkeley (1980)
• M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) 5–99 (1983)
• A Grothendieck, Sketch of a Programme Available at \setbox0\makeatletter\@url http://people.math.jussieu.fr/~leila/grothendieckcircle/mathtexts.php {\unhbox0
• P B Kronheimer, T S Mrowka, Scalar curvature and the Thurston norm, Math. Res. Lett. 4 (1997) 931–937
• S Mac Lane, Homology, Classics in Math., Springer, Berlin (1995) Reprint of the 1975 edition
• S Matsumoto, Some remarks on foliated $S\sp 1$ bundles, Invent. Math. 90 (1987) 343–358
• J Milnor, A concluding amusement: symmetry breaking, from: “Collected papers of John Milnor, III: Differential topology”, Amer. Math. Soc. (2007) xvi+343
• P Ozsváth, Z Szabó, Link Floer homology and the Thurston norm, J. Amer. Math. Soc. 21 (2008) 671–709
• P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. $(2)$ 17 (1978) 555–565
• W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
• W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 59 (1986) i–vi and 99–130