Geometric Intersection Number and Analogues of the Curve Complex for Free Groups

For the free group FN of finite rank N ≥ 2 we construct a canonical Bonahon-type, continuous and Out(FN )-invariant geometric intersection form 〈 , 〉 : cv(FN ) × Curr(FN ) → R≥0. Here cv(FN ) is the closure of unprojectivized Culler-Vogtmann’s 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 R-trees. The projectivization of cv(FN ) provides a free group analogue of Thurston’s compactification of the 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.