  We discuss in this article a property of action of groups by isometries
called "well displacing". An action is said to be well displacing, if the
displacement function is equivalent to the the displacement function for the
action on the Cayley graph. We relate this property with the fact that orbit
maps are quasi-isometric embeddings. We first describe countrexamples that
shows this two notions are unrelated in general. On the other hand we explain
that for a certain class of groups -- in particular hyperbolic groups -- these
two properties are equivalent. In the course of our discussion, we introduce an
intrinsic property of the group -- that we called the U-property -- which says
quantitatively how the norm an element is controlled by the translation length
of finitely many related conjugacy classes. This property play a central role
in our discussion.
