We investigate diagonal actions of Polish groups and the related
intersection operator on closed subgroups of the acting group. The
Borelness of the diagonal orbit equivalence relation is characterized
and is shown to be connected with the Borelness of the intersection
operator. We also consider relatively tame Polish groups and give a
characterization of them in the class of countable products of
countable abelian groups. Finally an example of a logic action is
considered and its complexity in the Borel reducbility hierarchy
determined.