Measure equivalence invariants and examples

Measure Equivalence (ME) is an equivalence relation on groups that was first defined by Gromov in analogy to the notions of topological equivalence and quasi-isometry.
We give a short survey of ME, meant to illuminate some of the landscape of the nonamenable world and convey some interesting questions in the area. We begin by describing some of the classical antirigidity and rigidity results in the setting and introduce several invariants of ME. We then describe some recent results in the setting of wreath product groups.