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

Kapovich, Ilya¹ ; Lustig, Martin²

¹ Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA
² Mathématiques (LATP), Université Paul Cézanne - Aix Marseille III, ave Escadrille Normandie-Niémen, 13397 Marseille 20, France

Geometry & topology, Tome 13 (2009) no. 3, pp. 1805-1833.

Résumé

For the free group $F_N$ of finite rank $N\ge 2$ we construct a canonical Bonahon-type, continuous and $Out\left(F_N\right)$–invariant geometric intersection form

Here $\overline{cv}\left(F_N\right)$ is the closure of unprojectivized Culler–Vogtmann Outer space $cv\left(F_N\right)$ in the equivariant Gromov–Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\overline{cv}\left(F_N\right)$ consists of all very small minimal isometric actions of $F_N$ on $ℝ$–trees. The projectivization of $\overline{cv}\left(F_N\right)$ 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.

Détail

DOI : 10.2140/gt.2009.13.1805

Keywords: free group, Outer space, geodesic current, curve complex

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