We prove that Connes' Embedding Conjecture holds for the von Neumann algebras of sofic groups, that is sofic groups are hyperlinear. Hence we provide some new examples of hyperlinearity. We also show that the Determinant Conjecture holds for sofic groups as well. We introduce the notion of essentially free actions and amenable actions and study their properties.

We prove that if L is a finite simple group of Lie type and A a set of generators of L, then A grows i.e |A 3 | > |A| 1+ε where ε depends only on the Lie rank of L, or A 3 = L. This implies that for a family of simple groups L of Lie type the diameter of any Cayley graph is polylogarithmic in |L|. We also obtain some new families of expanders.We also prove the following partial extension. Let G be a subgroup of GL(n, p), p a prime, and S a symmetric set of generators of G satisfying |S 3 | ≤ K|S| for some K. Then G has two normal subgroups H ≥ P such that H/P is soluble, P is contained in S 6 and S is covered by K c cosets of H where c depends on n. We obtain results of similar flavour for sets generating infinite subgroups of GL(n, F), F an arbitrary field.

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Answering some queries of Weiss [5], we prove that the free product and amenable extensions of sofic groups are sofic as well, and give an example of a finitely generated sofic group that is not residually amenable.