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

TL;DR

This paper constructs a canonical intersection form for free groups that parallels Thurston's intersection form for surfaces, and demonstrates that certain free group analogues of the curve complex have infinite diameter.

Contribution

It introduces a new geometric intersection form for free groups that is continuous, invariant, and analogous to Thurston's form, extending the analogy between free groups and surface theory.

Findings

The intersection form is continuous and Out(F_N)-invariant.

The free group analogues of the curve complex have infinite diameter.

The form provides a new tool for studying free group actions on R-trees.

Abstract

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.