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

TL;DR

This paper investigates the relationship between currents and laminations in free groups, revealing complexities beyond the surface case of real trees dual to measured geodesic laminations.

Contribution

It extends the theory of laminations for free groups by analyzing the connection with currents and highlighting new complexities.

Findings

Currents and laminations are more intricately linked in free groups than in surface cases.

The dual $ $-tree metrics exhibit increased complexity in the free group setting.

The paper advances understanding of the structure of free group laminations.

Abstract

This is the third of a series of three articles where we introduce laminations for the free-groups. We explore here the link between currents and laminations and prove that the situation is more complicated than in the surface case of real tree dual to a measured geodesic lamination.