Spectral rigidity for primitive elements of FN

Abstract

Abstract Two trees in the boundary of outer space are said to be primitive-equivalent whenever their translation length functions are equal in restriction to the set of primitive elements of FN . 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 in [`The horoboundary of outer space, and growth under random automorphisms', preprint (2014)]. In proving this, 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 FN -equivariant map between the metric completion of any two minimal, very small FN -trees is equal to the supremal ratio between the translation lengths of the elements of FN in these trees. We also provide approximation results for trees in the boundary of outer space.