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Groups of Intermediate Growth and Grigorchuk's Group Groups of Intermediate Growth and Grigorchuk's Group Eilidh McKemmie supervised by Panos Papazoglou This was submitted as a 2CD Dissertation for FHS Mathematics and Computer Science, Oxford University, Hilary Term 2015 Abstract In 1983, Rostislav Grigorchuk [6] discovered the first known example of a group of intermediate growth. We construct Grigorchuk's group and show that it is an infinite 2-group. We also show that it has intermediate growth, and discuss bounds on the growth. Acknowledgements I would like to thank my supervisor Panos Papazoglou for his guidance and support, and Rostislav Grigorchuk for sending me copies of his papers. ii Contents 1 Motivation for studying group growth 1 2 Definitions and useful facts about group growth 2 3 Definition of Grigorchuk's group 5 4 Some properties of Grigorchuk's group 9 5 Grigorchuk's group has intermediate growth 21 5.1 The growth is not polynomial . 21 5.2 The growth is not exponential . 25 6 Bounding the growth of Grigorchuk's group 32 6.1 Lower bound . 32 6.1.1 Discussion of a possible idea for improving the lower bound . 39 6.2 Upper bound . 41 7 Concluding Remarks 42 iii iv 1 Motivation for studying group growth Given a finitely generated group G with finite generating set S , we can see G as a set of words over the alphabet S . Defining a weight on the elements of S , we can assign a length to every group element g which is the minimal possible sum of the weights of letters in a word which represents g. We are interested in the set of all elements in G with length bounded above by some number n. This set is exactly the ball of words of radius n around the identity in the Cayley graph of G with respect to S where the length of an edge labelled by s is the weight of s. The growth of G with respect to S and our chosen weight is, roughly speaking, the growth rate of the volume of this ball of words as we let the radius increase. We define an equivalence relation on growth functions such that, no matter how we weight group generators, the growth of G always falls into the same equivalence class. We can then classify the growth of any finitely generated group without having to specify the generating set or the weights we are working with. We should note that using weights to define growth is not standard. Usually we would simply assign a value of 1 to every generator. However, we will show that different weights can give us more precise information about the growth of a group, so a definition of group growth using weights will be useful. The historical motivation for studying the growth of finitely generated groups lies in differential geometry. In 1955, Svarcˇ showed in [14] (as discussed in [7]) that the growth rate of the volume of a ball in the universal cover of a compact Riemannian manifold is equivalent to the growth of the fundamental group of that manifold. Mil- nor and Wolf [12, 15] also noted a relationship between the curvature of a compact Riemannian manifold and the growth of its fundamental group: Milnor showed that the fundamental group of a compact manifold of negative curvature has exponential growth, and (as discussed by Wolf in [15]) that bounds on the curvature of a Rie- mannian manifold result in bounds on the growth of the fundamental group of the manifold. In 1968, Milnor posed a question in [13] asking whether there are any groups whose growth is intermediate, that is, not equivalent to any polynomial or exponential. The question was answered by Grigorchuk in [5, 6] in 1983 with the construction of un- countably many finitely generated groups of intermediate growth. We will construct a group of intermediate growth often called the first Grigorchuk group or simply Grigorchuk's group. We will discuss some of the group's properties and show it has intermediate growth. Grigorchuk originally conjectured that the growth p of the group was equivalent to e n , a conjecture which was disproved in 2000 when Leonov and Bartholdi [10, 1] both found that Grigorchuk's group grows strictly more p quickly than e n . We will discuss various bounds on the growth of Grigorchuk's group, and finally note that the exact nature of the growth of Grigorchuk's group is unknown. 1 2 Definitions and useful facts about group growth We begin by defining our terms and notation. Notation 2.1. Throughout this work, we will be using log for the natural logarithm and e for the identity element of a group. We also say that the set of natural numbers N does not contain 0. Definition 2.2 (Alphabets and words). An alphabet is a non-empty set whose ele- ments we sometimes call letters. A word over an alphabet Σ is a finite sequence of letters (s1; : : : ; sk) where si 2 Σ. We usually omit the parentheses and commas and 0 write s1 : : : sk for the word (s1; : : : ; sk). If w; w are words over the alphabet Σ then we denote their concatenation by ww0 . For n 2 N we let the word wn denote the con- catenation of the word w with itself n times. We let Σm denote the set of all words on Σ with exactly m letters, and Σ∗ denote the set of all words on Σ. The empty word is denoted . For this section, let G be a finitely generated group with finite generating set S , and define S−1 = fs−1 : s 2 Sg. Then all elements of G can be represented by words over the alphabet S [ S−1 in the obvious way. There may be two words which are not equal as words but which represent the same group element. The identity element e is represented by the empty word . Definition 2.3 (Weights, word lengths and group element lengths). A weight δ on S −1 −1 is a function δ : S [ S [ fg ! R≥0 such that δ(s) = δ(s ) > 0 for all s 2 S and δ() = 0. −1 Define the length jwj of the word w = s1 ··· sn on the alphabet S [ S to be jwj = n, the number of letters in the word. Define the weight δ(w) of the word w = s1 ··· sn to be δ(w) = δ(s1) + ··· + δ(sn). For g 2 G, the length of g is lδ(g) = minfδ(w): w is a word representing gg. That is, lδ(g) is the minimum weight of a word representing g over the alphabet S . We write l(g) for lδ(g) where it is obvious which weight function we are using. Definition 2.4 (Group growth). Let δ be a weight on S . Define the ball of words of radius n around the identity to be Bδ(n) := fg 2 G : lδ(g) ≤ ng for n 2 R≥0 . The growth function γδ : R≥0 ! N is defined by γδ(n) := jBδ(n)j. We write B = Bδ and γ = γδ where there is no ambiguity. Note that B(0) = feg and so γ(0) = 1. It is quite unusual to define growth using weights, and for the most part, we will be using the standard weight defined by δ(s) = 1 for all s 2 S [ S−1 . However, different weights will be useful to us when trying to achieve tight bounds on the growth of Grigorchuk's group. We define an equivalence relation on functions so that we can classify the growth functions of groups. 2 0 Definition 2.5. Given two functions γ; γ : R≥0 ! N, say that γ does not grow more quickly than γ0 , written γ γ0 , if there exist constants α; C > 0 such that γ(n) ≤ Cγ0(αn) for all n > 0. We say γ and γ0 are equivalent, written γ ∼ γ0 , if both γ γ0 and γ0 γ . Proposition 2.6. ∼ is an equivalence relation. Proof. For reflexivity, we obviously have γ(n) γ(n), so γ(n) ∼ γ(n). For symmetry, note that if γ(n) ∼ γ0(n) then γ0(n) γ(n) and γ(n) γ0(n), and so γ0(n) ∼ γ(n). Transitivity of ∼ relies on the transitivity of , which we will show first: if γ(n) γ0(n) and γ0(n) γ00(n) then there exist constants C; D; α; β > 0 such that γ(n) ≤ Cγ0(αn) and γ0(n) ≤ Dγ00(βn), so γ(n) ≤ CDγ00(αβn). Therefore γ(n) γ00(n). So if γ0(n) ∼ γ(n) and γ0(n) ∼ γ00(n) then γ(n) γ0(n) γ00(n) and γ00(n) γ0(n) γ(n), which gives us that γ ∼ γ00(n) and so ∼ is transitive. Proposition 2.7 (The growth function is well-defined, [2, Lemma 2]). If S1 and S2 are two finite generating sets of the group G with weights δ1 on S1 and δ2 on S2 then γδ1 ∼ γδ2 . Proof. It is enough to show that γδ1 γδ2 . The other direction follows by symmetry. Let lδ2 (s) M := max : s 2 S1 > 0: δ1(s) Let w = s1 ··· sk (where si 2 S1 ) be a word representing g 2 G such that δ1(w) = lδ1 (g), ∗ Pk that is, w is a minimal-weight word in S1 representing g. Then lδ1 (g) = i=1 δ1(si). lδ2 (si) Note that, for i = 1; : : : ; k, we have M ≥ and so Mδ1(si) ≥ lδ2 (si). Therefore δ1(si) k k X X lδ2 (g) ≤ lδ2 (si) ≤ Mδ1(si) = Mlδ1 (g): i=1 i=1 If g 2 Bδ1 (n) then lδ1 (g) ≤ n, so lδ2 (g) ≤ Mlδ1 (g) ≤ Mn, giving us that g 2 Bδ2 (Mn).
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