There exist finitely generated simple groups that are infinite and amenable

You need to know: Group, finite and infinite groups, isomorphic and nonisomorphic groups, simple group, finitely generated group.

Background: A group G is called amenable if there is a map from subsets of G to such that (i) , (ii) whenever , and (iii) for all , where .

The Theorem: On 10th April 2012, Kate Juschenko and Nicolas Monod submitted to arxiv a paper in which they proved the existence of finitely generated simple groups that are infinite and amenable.

Short context: Amenable groups were introduced by von Neumann in 1929 in relation to Banach–Tarski paradox, see here, and are extensively studied since that. However, before 2012, it was an open question if there exists any finitely generated simple group that is infinite and amenable. The Theorem proves that such groups exist. Moreover, Juschenko and Monod proved that there are infinitely many (in fact uncountably many) nonisomorphic such groups. See here for even more general result proved in a later work.

Links: Free arxiv version of the original paper is here, journal version is here.

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