 It is conjectured that the limit

exists but even this remains a major open problem.Grigorchuk's group was also the first example of a group that is amenable but not elementary amenable, thus answering a problem posed by Mahlon Day in 1957.Originally, Grigorchuk's group "G" was constructed as a group of Lebesgue-measure-preserving transformations on the unit interval, but subsequently simpler descriptions of "G" were found and it is now usually presented as a group of automorphisms of the infinite regular binary rooted tree.