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Concept
Grigorchuk group
Summary
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth. The group was originally constructed by Grigorchuk in a 1980 paper and he then proved in a 1984 paper that this group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968. The Grigorchuk group remains a key object of study in geometric group theory, particularly in the study of the so-called branch groups and automata groups, and it has important connections with the theory of iterated monodromy groups.
The growth of a finitely generated group measures the asymptotics, as of the size of an n-ball in the Cayley graph of the group (that is, the number of elements of G that can be expressed as words of length at most n in the generating set of G). The study of growth rates of finitely generated groups goes back to the 1950s and is motivated in part by the notion of volume entropy (that is, the growth rate of the volume of balls) in the universal covering space of a compact Riemannian manifold in differential geometry. It is obvious that the growth rate of a finitely generated group is at most exponential and it was also understood early on that finitely generated nilpotent groups have polynomial growth. In 1968 John Milnor posed a question about the existence of a finitely generated group of intermediate growth, that is, faster than any polynomial function and slower than any exponential function. An important result in the subject is Gromov's theorem on groups of polynomial growth, obtained by Gromov in 1981, which shows that a finitely generated group has polynomial growth if and only if this group has a nilpotent subgroup of finite index.
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