ABSTRACT. We prove that the commutator is stable in permutations endowed with the Hamming distance, that is, two permutations that almost commute are near two commuting permutations.Our result extends to k-tuples of almost commuting permutations, for any given k, and allows restrictions, for instance, to even permutations.
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.
ABSTRACT. We introduce and systematically study linear sofic groups and linear sofic algebras. This generalizes amenable and LEF groups and algebras. We prove that a group is linear sofic if and only if its group algebra is linear sofic. We show that linear soficity for groups is a priori weaker than soficity but stronger than weak soficity. We also provide an alternative proof of a result of Elek and Szabo which states that sofic groups satisfy Kaplansky's direct finiteness conjecture. These group properties can be stated in elementary algebraic terms, as approximation properties, or in the language of ultraproducts, as the existence of an embedding in
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