$\mathbb{R}$-trees, dual laminations, and compact systems of partial isometries

Abstract

Let $\FN$ be a free group of finite rank $N\ge 2$, and let $T$ be an $\mathbb{R}$-tree with a very small, minimal action of $\FN$ with dense orbits. For any basis $\CA$ of $\FN$ there exists a {\em heart} $K_{\CA} \subset \bar T$ (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries $a_{i} : K_{\CA} \cap a_{i} K_{\CA} \to a_{i}\inv K_{\CA} \cap K_{\CA}$, for each $a_{i} \in \CA$, defines a tree $T_{(K_{\CA}, \CA)}$ which contains an isometric copy of $T$ as minimal subtree.

minor update

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 I: Algebraic laminations

Cited by in corpus (4)

• The boundary of the complex of free factors
• Algebraic laminations for free products and arational trees
• Ergodic currents dual to a real tree
• An algorithm to identify automorphisms which arise from self-induced interval exchange transformations