Examples of groups that are measure equivalent to the free group

Measure equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field has grown considerably over the past few years, developing tools to distinguish between different ME classes of countable groups. On the other hand, constructions of ME 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 ME for countable groups.