Geometric Intersection Number and analogues of the Curve Complex for free groups

Details Geometric Intersection Number and analogues of the Curve Complex for free groups Geometric Intersection Number and analogues of the Curve Complex for free groups

2007-11-24

arxiv.org/abs/0711.3806v4

Geometry and Topology, vol. 13 (2009), no. 3, pp. 1805-1833

Mathematics

Group Theory

Revised version, to appear in Geometry & Topology

Scientific paper

10.2140/gt.2009.13.1805

For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \emph{geometric intersection form} \[ <, >: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \] Here $\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\bar{cv}(F_N)$ consists of all \emph{very small} minimal isometric actions of $F_N$ on $\mathbb R$-trees. The projectivization of $\bar{cv}(F_N)$ provides a free group analogue of Thurston's compactification of the Teichm\"uller space. As an application, using the \emph{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.

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