Growth and Amenability of the Grigorchuk Group Chris Kennedy

In this note, we discuss the amenability of the first Grigorchuk group by appealing to its growth properties. In particular, we show that the first Grigorchuk group Γ has intermediate growth, hence providing an exotic example of an with non-polynomial growth. We also note some other interesting properties of Γ without proof, as well as a few examples of important groups constructed in the same way. To fix some common notation: for a group G generated by a finite set S we denote by `S the word metric on G with respect to S; S will almost always be implicit, so we will just write `. Additionally, for g ∈ G, we write `(g) = `(g, 1), the distance from g to the identity; this is the minimum number of letters in S required to represent g.

1. Basic Notions: Wreath Recursion and Action on Trees The workings of the Grigorchuk group are conveniently expressed using the concept of wreath recursion, which can be thought of as an algebraic structure for the action of groups on trees. Other approaches include finite-state automata (used extensively in [1]) and tableaux. ∗ Fix a finite set X = {x1, . . . , xd}, and let X be the tree on X, i.e. a tree whose vertices are strings in X, and in which vertices v, w are connected if and only if v = wx or w = vx for some x ∈ X (we include the empty string ∅ as the root vertex). Denote by Xk the kth level of X∗; that is, the set of words of length k. We call a map φ : X∗ → X∗ an automorphism if it is a (rooted) graph isomorphism; that is, it preserves levels (φ(Xk) = Xk) and adjacency (if v, w are adjacent, so are φ(v) and φ(w)). The group Aut X∗ of automorphisms of X∗ is uncountable and comes equipped with a profinite topology; but we will be more interested in countable of Aut X∗. For any vertex v ∈ X∗, note that vX∗ is isomorphic to X∗, so if g ∈ Aut X∗, it induces an ∗ ∗ isomorphism vX → g(v)X . This induces the restriction of g in v, denoted g|v, which is also an ∗ automorphism of X and is defined so that for any word w, g(vw) = g(v)g|v(w).

Definition: For an arbitrary group G, the permutational wreath product S(X) o G is the semi- X X direct product S(X) n G , where S(X) acts on G by permutation of the factors.

Since S(X) o G has the structure of S(X) × Gd, we may write any element of S(X) o G as σ(g1, . . . , gd) for some σ ∈ S(X) and g1, . . . , gd ∈ G. With this notation, multiplication obeys the rule

α(g1, . . . , gd) · β(h1, . . . , hd) = αβ(gβ(1)h1, . . . , gβ(d)hd)

Proposition: The group Aut X∗ has an isomorphism ψ : Aut X∗ → S(X) o Aut X∗, called the wreath homomorphism or wreath recursion, defined by ψ(g) = α(g|x1 , . . . , g|xd ), where α acts on X1 in the same way as g. Proof : Exercise.

1 2

Any G of Aut X∗ for which ψ as in the proposition exists such that ψ(G) is a subgroup of S(X)oG is said to be self-similar. We will use wreath recursion to notate elements in self-similar groups, and in particular Γ, by abusively identifying g with ψ(g).

2. Construction of Γ We base the construction on the exposition in §1.6 of [1]. The corresponding (quite similar) construction in [2] takes place in §VIII.B.

Let X = {0, 1}. We define Γ ≤ Aut X∗ as the group generated by the four automorphisms a, b, c, d, whose actions on the tree X∗ are pictured above (a vertex is marked with an arc if and only if that automorphism exchanges the subtrees under the vertex) (image taken from [1]). The generators are defined recursively by

a(0w) = 1w a(1w) = 0w b(0w) = 0a(w) b(1w) = 1c(w) c(0w) = 0a(w) c(1w) = 1d(w) d(0w) = 0w d(1w) = 1b(w) In the language of wreath recursion, we have a = σ(1, 1), b = (a, c), c = (a, d), and d = (1, b), where σ is the nontrivial permutation in S(X).

Remark: There is a more general notion of Grigorchuk group, in which for any infinite string ω w ∈ {0, 1, 2} , one constructs generators bw, cw, dw satisfying wreath recursions similar to those above and looks at Γw = ha, bw, cw, dwi. It turns out that two such groups Γw1 and Γw2 (with nei- ther w1 nor w2 eventually constant) are isomorphic if and only if w2 = σ(w1) for some σ ∈ S(3) (where σ is applied to each letter in w1). There are thus uncountably many Grigorchuk groups. For more, see §2.10.5 in [1].

As a simple example of arithmetic in Γ, one can check that a(g, h)a = (h, g). For a more inter- esting example, let us examine the first stabilizer subgroup, St(1) = {g ∈ G : g(0) = 0, g(1) = 1}. Any element of St(1) can be written as g = (g|0, g|1), and we may define homomorphisms φi : St(1) → Γ by φi(g) = g|i for i = 0, 1. In fact, we have: 3

Lemma: The maps φi : St(1) → Γ defined above are surjective, and hence Γ is infinite. Proof : We just need to show that a, b, c, d are in the images of φ0 and φ1. Note that aba = (c, a), aca = (d, a), and ada = (b, 1), so that b, c, d ∈ φ0(St(1)), as the images of ada, aba, and aca respectively. This also shows that a ∈ φ1(St(1)), and together with the usual presentations of b, c, d given above, we get that φ0 and φ1 are surjective. We have just shown that a nontrivial subgroup of Γ surjects onto Γ, so Γ is infinite. 

One can in fact glean more from the above argument, namely that Γ is commensurable with Γ × Γ (that is, one is isomorphic to a finite-index subgroup of the other). Specifically, the image of St(1) in Γ × Γ under the map φ0 × φ1 has index 8. This fact implies that the growth function β(k) for Γ is (roughly) of the same as that of Γ × Γ, which cannot hold if Γ has polynomial growth. Hence β(k) grows faster than any polynomial. This is made precise in §VIII.62-63 of [2].

Lemma: The subset {1, b, c, d} ⊂ Γ is a subgroup isomorphic to the Klein group Z2 × Z2. Proof : Exercise.

With the above lemma, it is clear that Γ is a quotient of the free product Z2∗(Z2×Z2), meaning that every element of Γ can be written as1as2 ··· ask for some k, where all the si ∈ {b, c, d} (possibly without the a at the beginning, or with an a at the end). One can show (§VIII.E of [2]) using this fact that the word problem is solvable in Γ, but also that Γ is not finitely presentable. If g ∈ St(1), then we can conjugate the representation above with a or b, c, d as necessary to put it in the form as1as2 ··· ask (since St(1) is normal, conjugation keeps the element in St(1), and does not change its order, or its length enough to matter). This word must have an even number of a’s, and hence k is even. This allows us to write

g = (as1a)s2(as3a)s4 ··· (ask−1a)sk = (g|0, g|1) where we have already shown that asja ∈ {b, c, d} × {1, a} and si ∈ {a, 1} × {b, c, d} for all i, j. Hence g|0 and g|1 each have length at most k/2. Induction on length of g as well as some casework for g 6∈ St(1) eventually shows that Γ is 2n a 2-group, i.e. that for any g ∈ Γ, there is an n ∈ N such that g = 1. Thus Γ resolves the generalized Burnside problem on the existence of infinite, finitely generated torsion groups. However, Γ does not have finite exponent; one can find elements of Γ with any order 2n. So Γ does not do anything for the (classical?) Burnside problem, which asks whether a finitely generated group with finite exponent is finite. The answer to this question is also no, but the resulting groups are not amenable (unlike Γ). For a prime p ≥ 3, using the alphabet X = {1, 2, . . . , p}, one can construct a finitely generated, infinite p-group relatively simply using the following generators. Let σ be the cyclic permutation 1 7→ 2 7→ · · · 7→ p 7→ 1 in S(X). We let a = σ(1,..., 1) and recursively t = (a, a−1, 1, 1,..., 1, t). Then Gp = ha, ti is called the Gupta-Sidki p-group, and has the desired properties.

3. Growth Properties We have already proved that Γ does not have polynomial growth (there are other ways to do so; for example, show that Γ is not virtually nilpotent and apply Gromov’s theorem). We now show that, on the other hand, Γ also does not have exponential growth, and thus provides an example of a finitely generated group with intermediate growth. 4

From the argument at the end of the previous section, we get that for any element g ∈ St(1), `(g) + 1 `(g) + 1 `(g| ) ≤ and `(g| ) ≤ 0 2 1 2 Repeating this analysis over the first three levels (an arduous task), one gets:

X 3 Lemma: (Contraction) For g ∈ St(3), we have `(g| ) ≤ `(g) + 8. v 4 v∈X3 Proof : §VIII.59 in [2].

The lemma above is a condition that allows us to use what [1] calls a “strong contraction” argument (see §6.12.7), which can in general be used to show that a group has subexponential growth. In this case, the lemma gives us enough information about the growth of St(3), a finite index (normal) subgroup of Γ, to get useful information about the growth of Γ.

Lemma: Let ∆ be a group generated by a finite set S, and ∆0 ≤ ∆ a subgroup of finite index n. Let β(k) = |Bk(1)| be the size of the ball of radius k (in word length) centered at the identity in ∆, and β0(k) = |Bk(1) ∩ ∆0. Then β(k) ≤ nβ(k + n − 1). Proof : We can choose coset representatives δ1, . . . , δn for ∆0 in ∆ such that `(δj) ≤ j − 1 for −1 1 ≤ j ≤ n. So any x ∈ ∆ can be represented as x = δj y for some integer j and y ∈ ∆0, so `(y) ≤ `(δj) + `(x) ≤ n − 1 + `(x). Hence for any length k = `(x), we can find such a y of length at most k + n − 1, meaning that

|{x ∈ ∆ : `(x) ≤ k}| ≤ n|{y ∈ ∆0 : `(y) ≤ k + n − 1}| which is what we wanted to show. 

Theorem: The group Γ has subexponential growth. pk Proof : Take ∆ = Γ and ∆0 = St(3) in the previous lemma, and let ω = limk→∞ β(k). Let  > 0 be arbitrary. By Fekete’s lemma, there is an integer k0 such that (ω − )k ≤ β(k) ≤ (ω + )k k for all k ≥ k0; thus β(k) ≤ β(k0)(ω + ) for all k. Now, by the contraction lemma, in St(3) we have X β0(k) ≤ β(k1) ··· β(k8) 3 where the sum is taken over all ki such that k1 + ··· + k8 ≤ 4 k + 8, thus representing all 3 7 combinations of words with total length at most 4 k + 8. Recall that [Γ : St(3)] = 2 and let 8 C = 128β(k0) ; then

7 X k1+···+k8 β(k) ≤ 2 β0(k + n − 1) ≤ C (ω + )

3 7 where the sum now runs over k1 + ··· + k8 ≤ 4 (k + 2 − 1) + 8. It is well-known that the number 3 P (k) of 8-uples satisfying k1 + ··· + k8 ≤ 4 k + 8 is of polynomial growth in k, so 3 (k+27−1)+8 3 k 3 (27−1)+8 β(k) ≤ CP (k)(ω + ) 4 = (ω + ) 4 · CP (k)(ω + ) 4

3 and, taking the kth root of both sides and letting k → ∞, we get ω ≤ (ω + ) 4 . Since  was arbitrary, this means ω = 1, which means β(k), and hence Γ, has subexponential growth.  5

3 Remark: The end of the above proof shows why the constant 4 is important in the contrac- tion lemma; the more general case of the above proof would thus rely on finding a finite-index subgroup for which the analogous constant is less than 1, and the argument would hold mutatis mutandis.

Corollary: Γ is amenable. [Proof: all groups of subexponential growth are amenable.]

There has been lots of work done to more precisely understand the growth rate of Γ, since the above analysis only shows that β(k) is of order ekα for some 0 < α < 1. It is not too hard to 1 show that α ≥ 2 , and the first upper bound less than 1 was log32 31. It is currently known that 0.5157 < α < 0.7674.

4. Amenability of Γ and others We have shown that Γ has subexponential growth, hence is amenable. A classical question about amenability is whether all amenable groups are part of the family EG, which is the closure of the class of finite and abelian groups under taking subgroups, quotients, extensions, and direct limits. The group Γ answers this question in the negative; it can be proved that Γ is not in EG. Generalizing the question, then, perhaps amenable groups are built from EG and groups of subexponential growth (once again allowing extensions, direct limits, etc.). This new class of amenable groups was dubbed SG. However, even this is not enough to encompass all amenable groups: on the alphabet {0, 1}, take a = σ(b, 1) and b = (a, 1) as defining wreath relations for a group H = ha, bi. This group turns out to be the iterated monodromy group of the polynomial z2 − 1, but was first investigated just as a three-state automaton. Bartholdi and Virag proved that H is amenable (using random walks and self-similarity), but it is not in the class SG. The group H is sometimes called the basilica group. For more information about H, see §5.2.2 (construction) and §6.12.1 (properties) of [1].

5. References 1. V. Nekrashevych, Self-Similar Groups. American Mathematical Society, 2005.

2. P. de la Harpe, Topics in Geometric . University of Chicago Press, 2000.