Grigorchuk group
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[1] and he then proved in a 1984 paper[2] that this group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968.
[3] 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[4] 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.
Prior to Grigorchuk's work, there were many results establishing growth dichotomy (that is, that the growth is always either polynomial or exponential) for various classes of finitely generated groups, such as linear groups, solvable groups,[5][6] etc.
Grigorchuk's group G was constructed in a 1980 paper of Rostislav Grigorchuk,[1] where he proved that this group is infinite, periodic and residually finite.
In a subsequent 1984 paper[2] Grigorchuk proved that this group has intermediate growth (this result was announced by Grigorchuk in 1983).
[7] More precisely, he proved that G has growth b(n) that is faster than
[9] It was conjectured that the limit and this remained a major open problem until the problem was resolved in 2020 by Anna Erschler and Tianyi Zheng[10] in which it was shown that the limit equals
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 Marsh Day in 1957.
[11] 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.
Recently important connections between this theory and complex dynamics, particularly the notion of iterated monodromy groups, have been uncovered in the work of Volodymyr Nekrashevych,[12] and others.
After Grigorchuk's 1984 paper, there were many subsequent extensions and generalizations.
[13][14][15][16] Although initially the Grigorchuk group was defined as a group of Lebesgue measure-preserving transformations of the unit interval, at present this group is usually given by its realization as a group of automorphisms of the infinite regular binary rooted tree T2.
The tree T2 is the set Σ* of all finite strings in the alphabet Σ = {0,1}, including the empty string ∅, which roots T2.
The group of all automorphisms Aut(T2) can thus be thought of as the group of all length-preserving permutations σ of Σ* that also respect the initial segment relation: whenever a string x is an initial segment of a string y then σ(x) is an initial segment of σ(y).
The Grigorchuk group G is the subgroup of Aut(T2) generated by four specific elements of Aut(T2) defined as follows (note that ∅ is fixed by any tree-automorphism):
Only the element a is defined explicitly; it swaps the child trees of ∅.
The elements b, c, and d are defined through a mutual recursion.
To understand the effect of the latter operations, consider the rightmost branch B of T2, which begins {∅, 1, 11, 111, ...}.
The original tree T2 can be obtained by rooting a tree isomorphic to T2 at each element of B; conversely, one can decompose T2 into isomorphic subtrees indexed by elements of
The operations b, c, and d all respect this decomposition: they fix each element of B and act as automorphisms on each indexed subtree.
Likewise, when c acts, it fixes only the subtrees of index ≡ 1 (mod 3); and d fixes those of index ≡ 0 (mod 3).
A compact notation for these operations is as follows: let the left (resp.
We write f = (g, h) to mean that f acts as g on TL and as h on TR.
The following are basic algebraic properties of the Grigorchuk group (see[17] for proofs):
