Geometry & Topology

Geometric intersection number and analogues of the curve complex for free groups

Ilya Kapovich and Martin Lustig

Full-text: Open access


For the free group FN of finite rank N2 we construct a canonical Bonahon-type, continuous and Out(FN)–invariant geometric intersection form

, : cv ¯ ( F N ) × Curr ( F N ) 0 .

Here cv¯(FN) is the closure of unprojectivized Culler–Vogtmann Outer space cv(FN) in the equivariant Gromov–Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv¯(FN) consists of all very small minimal isometric actions of FN on –trees. The projectivization of cv¯(FN) provides a free group analogue of Thurston’s compactification of Teichmüller space.

As an application, using the intersection graph determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.

Article information

Geom. Topol., Volume 13, Number 3 (2009), 1805-1833.

Received: 26 August 2008
Accepted: 6 November 2008
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]
Secondary: 57M99: None of the above, but in this section 37B99: None of the above, but in this section 37D99: None of the above, but in this section

free group Outer space geodesic current curve complex


Kapovich, Ilya; Lustig, Martin. Geometric intersection number and analogues of the curve complex for free groups. Geom. Topol. 13 (2009), no. 3, 1805--1833. doi:10.2140/gt.2009.13.1805.

Export citation


  • J Behrstock, M Bestvina, M Clay, Growth of intersection numbers for free group automorphisms
  • M Bestvina, M Feighn, Outer limits Available at \setbox0\makeatletter\@url {\unhbox0
  • M Bestvina, M Feighn, The topology at infinity of ${\rm Out}(F\sb n)$, Invent. Math. 140 (2000) 651–692
  • M Bestvina, M Feighn, M Handel, A hyperbolic $\Out(F_n)$–complex
  • M Bestvina, M Handel, Train tracks and automorphisms of free groups, Ann. of Math. $(2)$ 135 (1992) 1–51
  • F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. $(2)$ 124 (1986) 71–158
  • F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139–162
  • F Bonahon, Geodesic currents on negatively curved groups, from: “Arboreal group theory (Berkeley, CA, 1988)”, (R C Alperin, editor), Math. Sci. Res. Inst. Publ. 19, Springer, New York (1991) 143–168
  • M M Cohen, M Lustig, Very small group actions on ${\bf R}$–trees and Dehn twist automorphisms, Topology 34 (1995) 575–617
  • T Coulbois, A Hilion, M Lustig, $\Bbb R$–trees and laminations for free groups. I. Algebraic laminations, J. Lond. Math. Soc. $(2)$ 78 (2008) 723–736
  • T Coulbois, A Hilion, M Lustig, $\Bbb R$–trees and laminations for free groups. II. The dual lamination of an $\Bbb R$–tree, J. Lond. Math. Soc. $(2)$ 78 (2008) 737–754
  • T Coulbois, A Hilion, M Lustig, $\Bbb R$–trees and laminations for free groups. III. Currents and dual $\Bbb R$–tree metrics, J. Lond. Math. Soc. $(2)$ 78 (2008) 755–766
  • M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91–119
  • S Francaviglia, Geodesic currents and length compactness for automorphisms of free groups, Trans. Amer. Math. Soc. 361 (2009) 161–176
  • D Gaboriau, A Jaeger, G Levitt, M Lustig, An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 (1998) 425–452
  • E Ghys, P de la Harpe (editors), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math. 83, Birkhäuser, Boston (1990) Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988
  • V Guirardel, Dynamics of ${\rm Out}(F\sb n)$ on the boundary of outer space, Ann. Sci. École Norm. Sup. $(4)$ 33 (2000) 433–465
  • V Guirardel, Cœ ur et nombre d'intersection pour les actions de groupes sur les arbres, Ann. Sci. École Norm. Sup. $(4)$ 38 (2005) 847–888
  • U Hamenstädt, Subgroups of $\Out(F_n)$, Lecture at the workshop “Discrete groups and geometric structures", Kortrijk, Belgium (2008)
  • A Hatcher, K Vogtmann, The complex of free factors of a free group, Quart. J. Math. Oxford Ser. $(2)$ 49 (1998) 459–468
  • J Hempel, $3$–manifolds as viewed from the curve complex, Topology 40 (2001) 631–657
  • V Kaimanovich, I Kapovich, P Schupp, The subadditive ergodic theorem and generic stretching factors for free group automorphisms, Israel J. Math. 157 (2007) 1–46
  • I Kapovich, The frequency space of a free group, Internat. J. Algebra Comput. 15 (2005) 939–969
  • I Kapovich, Currents on free groups, from: “Topological and asymptotic aspects of group theory”, (R Grigorchuk, M Mihalik, M Sapir, Z Šuni\'k, editors), Contemp. Math. 394, Amer. Math. Soc. (2006) 149–176
  • I Kapovich, Clusters, currents, and Whitehead's algorithm, Experiment. Math. 16 (2007) 67–76
  • I Kapovich, M Lustig, Domains of proper discontinuity on the boundary of Outer space
  • I Kapovich, M Lustig, Intersection form, laminations and currents on free groups, to appear in Geom. Funct. Anal.
  • I Kapovich, M Lustig, Ping-pong and Outer space
  • I Kapovich, M Lustig, The actions of ${\rm Out}(F\sb k)$ on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility, Ergodic Theory Dynam. Systems 27 (2007) 827–847
  • I Kapovich, T Nagnibeda, The Patterson–Sullivan embedding and minimal volume entropy for outer space, Geom. Funct. Anal. 17 (2007) 1201–1236
  • G Levitt, M Lustig, Irreducible automorphisms of $F\sb n$ have north-south dynamics on compactified outer space, J. Inst. Math. Jussieu 2 (2003) 59–72
  • M Lustig, A generalized intersection form for free groups, Preprint (2004)
  • M Lustig, Conjugacy and centralizers for iwip automorphisms of free groups, from: “Geometric group theory”, (G N Arzhantseva, L Bartholdi, J Burillo, E Ventura, editors), Trends Math., Birkhäuser, Basel (2007) 197–224
  • R Martin, Nonuniquely ergodic foliations of thin type, measured currents and automorphisms of free groups, PhD thesis, UCLA (1995)
  • H A Masur, Y N Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • F Paulin, The Gromov topology on ${\bf R}$–trees, Topology Appl. 32 (1989) 197–221
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • H Zieschang, E Vogt, H-D Coldewey, Flächen und ebene diskontinuierliche Gruppen, Lecture Notes in Math. 122, Springer, Berlin (1970)