Spectral rigidity for primitive elements of $F_N$

TL;DR

This paper characterizes primitive-equivalence of trees in outer space boundary via translation length functions, extending White's theorem to boundary trees and exploring the horoboundary of outer space.

Contribution

It provides an explicit description of primitive-equivalence, extends White's theorem to boundary trees, and offers approximation results for trees in the boundary of outer space.

Findings

Primitive-equivalence is nontrivial.

Infimal Lipschitz constant equals the supremal ratio of translation lengths.

Extended White's theorem to boundary trees.

Abstract

Two trees in the boundary of outer space are said to be \emph{primitive-equivalent} whenever their translation length functions are equal in restriction to the set of primitive elements of $F_N$. We give an explicit description of this equivalence relation, showing in particular that it is nontrivial. This question is motivated by our description of the horoboundary of outer space for the Lipschitz metric. Along the proof, we extend a theorem due to White about the Lipschitz metric on outer space to trees in the boundary, showing that the infimal Lipschitz constant of an $F_N$-equivariant map between the metric completion of any two minimal, very small $F_N$-trees is equal to the supremal ratio between the translation lengths of the elements of $F_N$ in these trees. We also provide approximation results for trees in the boundary of outer space.