#### Volume 13, issue 3 (2009)

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

### Ilya Kapovich and Martin Lustig

Geometry & Topology 13 (2009) 1805–1833
##### Abstract

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

$〈\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}〉:\phantom{\rule{0.3em}{0ex}}\overline{cv}\left({F}_{N}\right)×Curr\left({F}_{N}\right)\to {ℝ}_{\ge 0}.$

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.

##### Keywords
free group, Outer space, geodesic current, curve complex
##### Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 57M99, 37B99, 37D99