Indecomposable $F_N$-trees and minimal laminations

TL;DR

This paper develops new techniques to analyze boundary actions in Outer Space, linking free indecomposable trees to minimal dual laminations, thus generalizing previous foundational results in the field.

Contribution

It introduces an inductive method inspired by Rauzy-Veech induction for boundary actions, connecting tree indecomposability with lamination minimality.

Findings

A tree in Outer space boundary is free and indecomposable iff its dual lamination is minimal.

Generalization of key results from BFH97 and KL11.

New adaptation of classical induction techniques for boundary analysis.

Abstract

We extend the techniques of [CH] to build an inductive procedure for studying actions in the boundary of the Culler-Vogtmann Outer Space, the main novelty being an adaptation of he classical Rauzy-Veech induction for studying actions of surface type. As an application, we prove that a tree in the boundary of Outer space is free and indecomposable if and only if its dual lamination is minimal up to diagonal leaves. Our main result generalizes [BFH97, Proposition 1.8] as well as the main result of [KL11].