  The notion of sofic equivalence relation was introduced by Gabor Elek and
Gabor Lippner. Their technics employ some graph theory. Here we define this
notion in a more operator algebraic context, starting from Connes' embedding
problem, and prove the equivalence of this two definitions. We introduce a
notion of sofic action for an arbitrary group and prove that amalgamated
product of sofic actions over amenable groups is again sofic. We also prove
that amalgamated product of sofic groups over an amenable subgroup is again
sofic.
