  Measure Equivalence (ME) is the measure theoretic counterpart of
quasi-isometry. This field grew considerably during the last years, developing
tools to distinguish between different ME classes of countable groups. On the
other hand, contructions of ME equivalent groups are very rare. We present a
new method, based on a notion of measurable free-factor, and we apply it to
exhibit a new family of groups that are measure equivalent to the free group.
We also present a quite extensive survey on results about Measure Equivalence
for countable groups.
