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

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

$$\langle ,\rangle :\overline{cv}\left(F_N\right)\times Curr\left(F_N\right)\to {ℝ}_{\ge 0}.$$

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.