$\mathbb{R}$-trees and laminations for free groups I: Algebraic laminations

Abstract

This paper is the first of a sequence of three papers, where the concept of an $\mathbb{R}$-tree dual to a measured geodesic lamination in a hyperbolic surface is generalized to arbitrary $\mathbb{R}$-trees provided with a (very small) action of the free group $F_N$ of finite rank $N\ge 2$ by isometries. Three different definitions are given and they are proved to be equivalent. We also describe the topology and Out$(F_N)$-action on the space of laminations.

19 pages

References in corpus (1)

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

Cited by in corpus (27)

• $\mathbb{R}$-trees and laminations for free groups II: The dual lamination of an $\mathbb{R}$-tree
• Geometric Intersection Number and analogues of the Curve Complex for free groups
• The boundary of the complex of free factors
• Intersection form, laminations and currents on free groups
• Algebraic laminations for free products and arational trees
• The Poisson boundary of $\text{Out}(F_N)$
• Ping-pong and Outer space
• Dynamics of Irreducible Endomorphisms of $F_n$
• Algebraic Ending Laminations and Quasiconvexity
• Stabilizers of $\mathbb{R}$-trees with free isometric actions of $F_N$
• $\mathbb{R}$-trees, dual laminations, and compact systems of partial isometries
• Dynamics of hyperbolic iwips
• Return words of linear involutions and fundamental groups
• Subset currents on free groups
• Hyperbolic extensions of free groups from atoroidal ping-pong
• Ergodic currents dual to a real tree
• Random trees in the boundary of Outer space
• Cannon-Thurston fibers for iwip automorphisms of $F_N$
• Graph towers, laminations and their invariant measures
• Conical limit points and the Cannon-Thurston map
• An algorithm to identify automorphisms which arise from self-induced interval exchange transformations
• Irreducible laminations for IWIP Automorphisms of free products and Centralisers
• Invariant laminations for irreducible automorphisms of free groups
• Poisson boundary of groups acting on real trees
• Planes in degenerate 3-manifolds
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$
• Limit sets of unfolding paths in Outer space