Geometry & Topology

Faces of the scl norm ball

Danny Calegari

Full-text: Open access


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 SB1H(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

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

Received: 22 July 2008
Revised: 19 January 2009
Accepted: 17 January 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20J05: Homological methods in group theory
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 20F12: Commutator calculus 55N35: Other homology theories 57M07: Topological methods in group theory

immersion surface free group bounded cohomology scl polyhedral norm rigidity hyperbolic structure rotation number


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

Export citation


  • 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 {\unhbox0
  • D Calegari, scl, Monograph Available at \setbox0\makeatletter\@url {\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 {\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 {\unhbox0
  • W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 59 (1986) i–vi and 99–130