Translation equivalence in free groups

Abstract

Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when two elements g , h g,h in a free group F F have the property that for every free isometric action of F F on an R \mathbb {R} -tree X X the translation lengths of g g and h h on X X are equal.