Gromov's Measure Equivalence and Rigidity of Higher Rank Lattices

In this paper the notion of Measure Equivalence (ME) of countable groups is studied.ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries.All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion.The main result of this paper is ME rigidity of higher rank lattices: any countable group which is ME to a lattice in a simple Lie group G of higher rank, is commensurable to a lattice in G.