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 $Out(F_N)$-invariant \emph{geometric intersection form} \[ <, >: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \] Here $\overline{cv}(F_N)$ 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 $\overline{cv}(F_N)$ consists of all \emph{very small} minimal isometric actions of $F_N$ on $\mathbb{R}$-trees. The projectivization of $\overline{cv}(F_N)$ provides a free group analogue of Thurston's compactification of the Teichmüller 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.

Revised version, to appear in Geometry & Topology

References in corpus (2)

• $\mathbb{R}$-trees and laminations for free groups II: The dual lamination of an $\mathbb{R}$-tree
• $\mathbb{R}$-trees and laminations for free groups III: Currents and dual $\mathbb{R}$-tree metrics

Cited by in corpus (28)

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• Hyperbolic extensions of free groups
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• Lines of minima in Outer space
• The co-surface graph and the geometry of hyperbolic free group extensions
• The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)
• Algebraic laminations for free products and arational trees
• The Poisson boundary of $\text{Out}(F_N)$
• Ping-pong and Outer space
• Central limit theorems for mapping class groups and $\text{Out}(F_N)$
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• North-South dynamics of hyperbolic free group automorphisms on the space of currents
• Generalized north-south dynamics on the space of geodesic currents
• Stabilizers of $\mathbb{R}$-trees with free isometric actions of $F_N$
• Dynamics of hyperbolic iwips
• A Birman exact sequence for Aut(F_n)
• Subset currents on free groups
• A train track directed random walk on $Out(F_r)$
• The mapping class group of connect sums of $S^2\times S^1$
• Ergodic currents dual to a real tree
• Fully irreducible Automorphisms of the Free Group via Dehn twisting in ${♯}_k(S^2\times S^1)$
• Uniform independence for Dehn twist automorphisms of a free group
• On the topological dimension of the Gromov boundaries of some hyperbolic $Out(F_N)$-graphs
• Spectral rigidity of automorphic orbits in free groups
• Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space
• An intersection functional on the space of subset currents on a free group
• Limit sets of unfolding paths in Outer space
• The Pants Graph of a Free Group