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

TL;DR

This paper explores the relationship between $ $-trees, dual laminations, and systems of partial isometries in free groups, introducing a compact subtree called the heart that encodes the dynamics of the group action.

Contribution

It constructs a canonical compact subtree called the heart for $ $-trees with dense orbits, linking group actions, laminations, and partial isometries in a novel way.

Findings

Existence of a canonical heart $K_{ ext{CA}}$ for $ $-trees with dense orbits.

The dynamical system of partial isometries on $K_{ ext{CA}}$ encodes the original tree.

Construction of a tree $T_{(K_{ ext{CA}}, ext{CA})}$ containing the original as a minimal subtree.

Abstract

Let $\FN$ be a free group of finite rank $N \geq 2$, and let $T$ be an $\R$-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.