Brigham Young University BYU ScholarsArchive Theses and Dissertations 2007-07-13 Infinite Product Group Keith G. Penrod Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mathematics Commons BYU ScholarsArchive Citation Penrod, Keith G., "Infinite Product Group" (2007). Theses and Dissertations. 976. https://scholarsarchive.byu.edu/etd/976 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. INFINITE PRODUCT GROUPS by Keith Penrod A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Brigham Young University August 2007 Copyright c 2007 Keith Penrod All Rights Reserved BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Keith Penrod This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date Gregory Conner, Chair Date James Cannon Date Eric Swenson BRIGHAM YOUNG UNIVERSITY As chair of the candidate’s graduate committee, I have read the thesis of Keith Penrod in its final form and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date Gregory Conner Chair, Graduate Committee Accepted for the Department Gregory Conner Graduate Coordinator Accepted for the College Thomas Sederberg, Associate Dean College of Physical and Mathematical Sciences ABSTRACT INFINITE PRODUCT GROUPS Keith Penrod Department of Mathematics Master of Science The theory of infinite multiplication has been studied in the case of the Hawaiian earring group, and has been seen to simplify the description of that group. In this paper we try to extend the theory of infinite multiplication to other groups and give a few examples of how this can be done. In particular, we discuss the theory as applied to symmetric groups and braid groups. We also give an equivalent definition to K. Eda’s infinitary product as the fundamental group of a modified wedge product. ACKNOWLEDGMENTS Many thanks to Dr. Conner for being an aggressive advisor and pushing me to finish this thesis on time. Thanks to Dr. Cannon and Dr. Humphries for their advice and their help with my presentation. Also to Dr. Swenson for joining my committee at the last minute. Table of Contents 1 Introduction 1 2 Hawaiian Earring 1 3 Infinite Product Structure 7 4 Classical Symmetric Group Theory 9 5 The Infinite Symmetric group SN 10 6 Finite Braid Groups 14 7 Infinite Braid Groups 17 8 Doubled Cone 22 9 Hawaiian Product 24 References 27 A Infinitary Product 28 vii List of Figures 1 The Hawaiian earring ........................... 2 2 All elements of S3 represented as braids. ................ 10 3 The first three elementary braids. .................... 15 4 The doubled cone over the Hawaiian earring. .............. 23 viii 1 Introduction In classical group theory, only finite multiplication is allowed. There are obvious reasons why this is the case, but this paper explores some of the possibilities of using infinite multiplication, how to make sense of it, and how it can in some cases simplify the description or “presentation” of a group. In Section 2 we discuss the properties of the Hawaiian earring group that will be useful for the remainder of the paper, particularly in Section 3 where the theory of infinite multiplication will be generalized. In Section 4 we introduce notation for permutations and permutation groups and we give results that are part of classical group theory and that will be extended to the infinite multiplication case in Section 5. We will give further examples of how infinite multiplication can be used by ex- amining braid groups. Finite braid groups and classical braid group theory will be introduced in Section 6 and the results for infinite braid groups and the theory of infinite multiplication will be given in Section 7. Finally, in Section 9, we discuss another way of thinking about infinite multipli- cation, which is based on the concept of Eda’s infinitary product. 2 Hawaiian Earring If we define C(r, p) to be a circle in the plane centered at p with radius of r, then we call the space [ E = C(1/n, (1/n, 0)) n∈N the Hawaiian earring and its fundamental group π1(E, (0, 0)) will be called the Hawai- ian earring group and will be denoted by H. The space E and the group H have been studied extensively by K. Eda, J. Cannon, G. Conner, and others. The results 1 Figure 1: The Hawaiian earring most pertinent to this paper will be found in [1]. We will cite those results here in this section and forgo the proofs. In the discussion of the Hawaiian earring we will talk about the inverse limit of free groups, so we define what we mean by that here. Definition 2.1. Let Fn denote the free group on the generators {a1, . , an}. For m ≥ n, define the map ϕm,n : Fm → Fn by ak if k ≤ n ϕm,n(ak) = . 1 if k > n Unless stated otherwise, lim F will mean the inverse limit of this system. ←− n Definition 2.2. Given a reduced word w ∈ Fn and a letter ai, we define the i-weight of w to be the number of times the letter ai appears in w—that is, the sum of the absolute values of the exponents—and we denote this wi(w). Proposition 2.3. The Hawaiian earring group H embeds in the inverse limit lim F . ←− n In particular, it can be identified with the elements of lim F with bounded weights for ←− n all letters. That is, (x , x ,...) ∈ lim F is in H if for each j, the sequence {w (x )} 1 2 ←− n j i i is bounded. By Proposition 2.3, we may also define the i-weight of a word in the Hawaiian earring group. That is, for h ∈ H and i ∈ N, wi(h) is defined to be max {wi(hj)} 2 th (where hj is the j coordinate of h). If we were to express h as a word in the letters {ai} rather than as a sequence of finite words, then we would see that ai would appear exactly wi(h) times. Just as in the case with free groups, we insist that we only deal with reduced words, which by [1] exist and are unique. We take a moment to note that elements of the inverse limit are really coherent sequences of words in free groups, but when we think of them as elements of the Hawaiian earring group it will be convenient to think of them as words themselves, rather than sequences of words. Thus the sequence (a1, a1, a1,...) will be denoted sim- ply a1 and will represent the homotopy class of a map that wraps one time around the outer-most loop of the Hawaiian earring. Similarly, the sequence (a1, a1a2, a1a2a3,...) will be denoted a1a2a3 ··· and will represent the homotopy class of a map that wraps once around each loop, in order from outer-most inward. Since we identify H with a subgroup of lim F , we will use these two different notations interchangeably. ←− n Also by Proposition 2.3 we see that there are natural maps ψn : H → Fn where ai i ≤ n ψn(ai) = 1 1 > n Q and indeed ψn is an epimorphism. It is noted that ψn = πn|H , where πn : Fi → Fn is the canonical projection. Therefore, we will call this map the projection onto Fn or the nth projection. Now we topologize H as follows: give each free group Fn the discrete topol- ogy, so that the Hawaiian earring group H inherits its topology from the product Q space Fn. It can be seen that a basis for this topology is given by the collection −1 {ψn (w) | n ∈ N, w ∈ Fn}. Furthermore, this is a metrizable space and the topology 3 is given by the metric 0 0 if w = w d(w, w0) = 0 0 1/n if ψn(w) 6= ψn(w ) and ψi(w) = ψi(w ) for i < n. To ensure that this metric is well-defined, we insist that both of w and w0 are reduced words. It is also important to note that Cannon and Conner have shown that for 0 0 w 6= w , there is an n such that ψn(w) 6= ψn(w ). The significance of this topology lies in the fact that we can define infinite products very similarly to the way analysts define series. Series are said to converge if the sequence of partial sums converges. Likewise, we will say that an infinite product of words in H converges if every sequence of partial products converges. In this case, we call the product legal and in the case that no sequence of partial products converge we call the product illegal. That is, the sequence {a1, a1a2, a1a2a3,...} converges to Q the point a1a2a3 ··· , so we say that the product ai is legal. However, we would first do well to define what an infinite product is. Definition 2.4. An infinite product on the Hawaiian earring group is a function f : Q → H. A subproduct f|A is pseudo-finite if f(a) = 1 for all but finitely many Y a ∈ A, and the value of the subproduct is ν(f|A) = f(a), multiplied in the order f(a)6=1 dictated by A.A subproduct chain will be a sequence of sets A1 ⊂ A2 ⊂ A3 ⊂ ..