  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 $g,h$ in a free
group $F$ have the property that for every free isometric action of $F$ on an
$\mathbb{R}$-tree $X$ the translation lengths of $g$ and $h$ on $X$ are equal.
We give a combinatorial characterization of this phenomenon, called translation
equivalence, in terms of Whitehead graphs and exhibit two difference sources of
it. The first source of translation equivalence comes from representation
theory and $SL_2$ trace identities. The second source comes from geometric
properties of groups acting on real trees and a certain power redistribution
trick. We also analyze to what extent these are applicable to the tree actions
of surface groups that occur in the Thurston compactification of the
Teichmuller space.
