  Let G be any finitely generated infinite group. Denote by K(G) the FC-centre
of G, i.e., the subgroup of all elements of G whose centralizers are of finite
index in G. Let QI(G) denote the group of quasi-isometries of G with respect to
word metric. We observe that the natural homomorphism from the group of
automorphisms of G to QI(G) is a monomorphism only if K(G) equals the centre
Z(G) of G. The converse holds if K(G)=Z(G) is torsion free. We apply this
criterion to many interesting classes of groups.
