  We show that a finitely generated group G which satisfies a certain condition
with respect to the Macaev norm is supramenable. The condition is equivalent to
the existence of quasicentral approximate unit with respect to the Macaev norm
relative to the left regular representation of the group and has been studied
by the author in connection with perturbation questions for Hilbert space
operators. The condition can be also viewed as an analogue with respect to the
Macaev norm of Yamasaki parabolicity. We also show that existence of
quasicentral approximate units relative to the Macaev norm for n-tuples of
operators is not preserved when taking tensor products.
