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On the Burnside Problem On the Burnside Problem Drew A. Hudec University of Chicago REU 2006 Abstract This paper will investigate the classical Burnside problem as it was originally proposed in 1902 and will focus on developing a counterexample to this conjecture involving automorphism groups on binary rooted trees. This example is more widely known as the first Grigorchuk group. I will introduce all the relevant ideas about binary rooted trees and their connection to the Grigorchuk group, so that this survey will be more self- contained. Although we will see that Burnside’s original conjecture is false in its full generality, I will also present a short historical overview of some related results that have since been attained. 1 Introduction to Burnside’s Problem Although having studied under the great algebraist Arthur Cayley while at Cambridge, William Burnside did not begin his mathematical career in this field. In fact, his university studies were focused in applied mathematics and later hydrodynamics. It was years later that his research shifted from applied mathematics to finite group theory, the subject that would later bring him fame. Today Burnside can be regarded as one of the most influential figures in the his- tory of this subject, after having drawn the interest of many mathematicians throughout the twentieth century with his proposed problems. Before stating his question, we will need a few preliminary definitions. Definition 1.1: Let G be a group and S ⊂ G. Let I = {H ≤ G | S ⊂ H}. T Then hSi = H∈I(H) is the subgroup of G generated by S. A group G is said to be finitely generated if it is generated by some finite subset S ⊂ G, ie. G = hSi. Definition 1.2: Let G be a group. An element g ∈ G has finite order if ∃n ∈ Z such that gn = 1. Definition 1.3: A group G is periodic if ∀g ∈ G, ∃n ∈ Z such that gn = 1, ie. Every element in G has finite order. 1 Now we are ready to state Burnside’s question: Is it the case that all finitely generated, periodic groups are finite? It may seem at first that this question should be answered in the affirmative. For there are only finitely many generators for the group, each of which has finite order. So this amounts to at most a finite number of distinct elements. Then all the finite products of the generators are also elements in the group, and thus these elements will each have finite order as well. It may seem as though we could somehow make the leap from this information to conclude that the group must then be finite. But this intuitive simplicity is only apparent. In the following sections we will define the first Gigorchuk group - one of the best known counterexamples to the Burnside problem - which will show that the answer to this question is no. 2 Graphs and Trees The definition of the Grigorchuk group involves permutations that act on cer- tain types of graphs. Formally, a graph G is a pair of sets G = (V, E) where V is a set of verticies and E is a set of edges. Edges are defined by pairs of verticies, and visualized by a line segment connecting them. Two verticies in a graph are said to be adjacent if the edge that they form is contained in E, and adjacent verticies are called neighbors. In a graph, a walk is a sequence of verticies v0, v1, ..., vk such that vi and vi+1 are adjacent for all i ∈ {0, ..., k − 1}. A cycle is a walk in which v0 = vk and there are no repeated verticies in the sequence beside v0 = vk. A graph is connected if there is a path between any two verticies in V . The following definition is necessary for our study of the Grigorchuk group. Definition 2.1: A tree is a connected graph with no cycles. 2.1 d-ary Rooted Trees A type of tree that will be of special interest to us will be the d-ary rooted tree T (d),(d > 1). These trees can be defined as follows: The set of verticies of T (d) consists of finite sequences of the integers {0, ..., d − 1}. The edges of T (d) connect pairs of verticies whose sequences differ in length by exactly one, and the shorter sequence can be obtained from the longer by deleting the last term. For example, vi = (1, 0, 0, 3, ) and vj = (1, 0, 0, 3, 2) would form an edge, whereas vl = (1, 0, 0, 2, 2) and vi would not. The root of the tree is defined to be the empty sequence, and we shall denote it by (∅). Every vertex (x1, ..., xk) 6= (∅) has exactly one parent, which is defined to be (x1, ..., xk−1) (Note that the root of T (d) has no parent). In the d-ary tree T (d), every vertex (x1, ..., xk) has d children, which are the verticies of the form (x1, ..., xk, xk+1) where xk+1 ∈ {0, ..., d − 1}. In the d-ary tree, we have a notion of levels, where 2 Figure 1: A binary rooted tree the kth level of T (d) is defined to be the subset of the vertex set V consisting of all the sequences with length k, denoted by L(d)(k). Thus, we can visualize the tree as having the root at the top and L(d)(k − 1) as being above L(d)(k). The Gigorchuk group is defined using binary rooted trees (trees in which every vertex has exactly two children, as in Figure 1), so as results are presented in the general setting of d-ary rooted trees, keep in mind that the binary case will be most important to us. 2.2 Automorphisms of T (d) Now that we have established the basic structure of T (d), we can consider func- tions acting on these trees. The first concept will be that of graph isomorphism. Definition 2.2: Let G1 = (V1,E1) and G2 = (V2,E2). A bijective function f : V1 → V2 is an isomorphism of graphs G1 and G2 if the function f preserves the structure of adjacency, ie. for any two adjacent verticies vi, vj ∈ G1, f(vi) and f(vj) are adjacent in G2. In the case that G1 = G2, we say that f is an automorphism. Note that an isomorphism of a graph maps verticies to verticies and edges to edges due to the preservation of adjacency. For any vertex x ∈ T (d), there (d) (d) exists a subtree Tx of T which contains all the verticies whose sequences begin with x. By concatenation of x with a vertex v ∈ T (d), we obtain a new (d) vertex xv in Tx . This defines an isomorphism: (d) (d) δx : T → Tx where v 7→ xv The automorphism group of all automorphisms on T (d) is denoted by G(d). Given any g ∈ G(d), g must fix the root. This is because the root has d neigh- bors, whereas every other vertex of T (d) has d+1 neighbors. So automorphisms (d) (d) (d) of T only permute the subtrees T0 , ..., Td−1 and fix (∅). 3 (d) (d) Definition 2.3: StG(d) (k) = {g ∈ G | ∀x ∈ L (k), g(x) = x} is the stabilizer subgroup of G(d) fixing L(d)(k). (d) Suppose we are given d automorphisms g0, g1, ..., gd−1 ∈ G . We can then (d) define a new automorphism g ∈ G that is in StG(d) (1) (i.e. g fixes the first level (d) of the tree), such that each subtree Tj is acted upon by gj, j ∈ {0, ..., d − 1}. th We may formalize this notion by noticing that the action of gj on the j subtree −1 is precisely the automorphism δjgjδj , where δj is as defined above. We can reverse this procedure to show that every element in the stabilizer of the first level, StG(d) (1), is of this form. Consider an automorphism g ∈ (d) StG(d) (1). Then, by definition, g fixes the first level of T . Furthermore, g (d) (d) acts on each of the subtrees T0 , ..., Td−1 as a collection of d automorphisms (d) −1 g0, ..., gd−1 ∈ G where each δjgjδj is the restriction of g to the subtree (d) Tj . Now we can define the following function, which is an isomorphism by the preceding discussion. (d) (d) ψ = (φ0, ..., φd−1): StG(d) (1) → G × · · · × G where g 7→ (g0, ..., gd−1) Although it is correct to write ψ(g) = (g0, ..., gd−1), for g ∈ StG(d) (1) we will agree to write g = (g0, ..., gd−1) instead. Now that a general theory of rooted d-ary trees has been established, we can focus our attention on the binary case and on the Burnside problem. 3 The First Grigorchuk Group It took over fifty years for a conclusive answer to Burnside’s question to be discovered. This was first shown in 1964 by Golod and Shafarevich who found a finitely generated infinite p-group. Years later in 1980, Rostislav Grigorchuk published a paper containing the construction of the first Grigorchuk group. This group is of special interest to mathematicians for a number of reasons including growth rates of words, the various interesting properties exhibited by its subgroups, and the close relationship it shares with the Burnside problem. This final aspect of the Grigorchuk group is what leads us to investigate its properties. The first Grigorchuk group is constructed out of elements in G(2), which are automorphisms of the binary rooted tree, T (2).
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