  The set SL(n) of n-string links has a monoid structure, given by the stacking
product. When considered up to concordance, SL(n) becomes a group, which is
known to be abelian only if n=1. In this paper, we consider two families of
equivalence relations which endow SL(n) with a group structure, namely the
C_k-equivalence introduced by Habiro in connection with finite type invariants
theory, and the C_k-concordance, which is generated by C_k-equivalence and
concordance. We investigate under which condition these groups are abelian, and
give applications to finite type invariants.
