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Lim - GROWTH RATE OF GROUPS CONG HAN LIM Abstract The...

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GROWTH RATE OF GROUPS CONG HAN LIM Abstract. The concept of growth rates in group theory has proven to be a potent tool for studying groups. For example, in 1981, Mikhail Gromov proved the set of finitely-generated virtually nilpotent groups are precisely the groups of polynomial growth. In this paper, we give an introduction to growth rate in group theory and provide many tools and examples to illustrate some of the methods used. We also construct the First Grigorchuk group and prove that it has intermediate growth. Contents Introduction 1 1. Word Length and Growth Rate 2 2. Exponential Growth 5 3. Polynomial Growth 8 The First Grigorchuk Group: A Group of Intermediate Growth 10 4. Rooted Binary Trees 10 5. Construction of the First Grigorchuk Group and Basic Properties 12 6. Superpolynomial Growth 14 7. Rewriting Rules 16 8. Subexponential Growth 17 9. Proof of Upper and Lower Bound Lemmas 19 Acknowledgments 21 References 21 Introduction Consider a group G and a generating set S = { s 1 , s 2 , . . . , s k } . We define the word length l S ( g ) of an element g G to be the length of the shortest decomposition g = s ± 1 i 1 · · · s ± 1 i n . The growth function of G with respect to the generating set S , denoted by γ S G , associates to each integer n 0 the number of elements g G such that l S ( g ) n . The growth rate of a group G is the asymptotic behavior of γ S G . It has been known since the 1960s that all finitely-generated groups have either polynomial, exponential or intermediate growth, but it was only in 1980 that Ros- tilav Grigorchuk constructed the first group proven to have intermediate growth. Also, while it has been known for many decades that all finitely-generated virtually nilpotent have polynomial growth, it was only in 1981 that the converse was proven by Mikhail Gromov. In this paper, we will introduce some of the concepts of growth and state some key results. We will also construct the First Grigorchuk group. The reader is assumed to be familiar with basic algebraic concepts. 1
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2 CONG HAN LIM The first half of this paper is designed to be more of an overview of some of the important concepts behind growth rates. The very first section focuses on introducing the concepts of growth, such as the properties of the word length (the shortest string of elements in the generating set needed to express an element) and the growth rate (the rate at which the number of elements increase with respect to an increase in the maximum word length allowed). We show that there are three classes of growth, namely polynomial, exponential and intermediate growth and lead up to the result that the growth rates of a finitely-generated group and any subgroup of finite index have to be equivalent. The next two sections tackle some of the tools behind understanding polynomial and exponential growth. We provide an analogue of the “Ping-Pong Lemma” for free semi-groups and also show that the group of upper-triangular matrices with 1’s along the diagonals and integers elsewhere has polynomial growth.
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