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Statistical Properties of Thompson's Group and Random Pseudo Manifolds Brigham Young University BYU ScholarsArchive Theses and Dissertations 2005-06-15 Statistical Properties of Thompson's Group and Random Pseudo Manifolds Benjamin M. Woodruff Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mathematics Commons BYU ScholarsArchive Citation Woodruff, Benjamin M., "Statistical Properties of Thompson's Group and Random Pseudo Manifolds" (2005). Theses and Dissertations. 568. https://scholarsarchive.byu.edu/etd/568 This Dissertation 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]. STATISTICAL PROPERTIES OF THOMPSON’S GROUP AND RANDOM PSEUDO MANIFOLDS by Benjamin M. Woodruff A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics Brigham Young University August 2005 Copyright c 2005 Benjamin M. Woodruff All Rights Reserved BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a dissertation submitted by Benjamin M. Woodruff This dissertation has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date James W. Cannon, Chair Date David Grant Wright Date Eric Lewis Swenson Date Tyler J. Jarvis Date Gregory R. Conner BRIGHAM YOUNG UNIVERSITY As chair of the candidate’s graduate committee, I have read the dissertation of Benjamin M. Woodruff 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 James W. Cannon Chair, Graduate Committee Accepted for the Department Tyler J. Jarvis Graduate Coordinator Accepted for the College G. Rex Bryce, Associate Dean College of Physical and Mathematical Sciences ABSTRACT STATISTICAL PROPERTIES OF THOMPSON’S GROUP AND RANDOM PSEUDO MANIFOLDS Benjamin M. Woodruff Department of Mathematics Doctor of Philosophy The first part of our work is a statistical and geometric study of properties of Thomp- son’s Group F. We enumerate the number of elements of F which are represented by a reduced pair of n-caret trees, and give asymptotic estimates. We also discuss the effects on word length and number of carets of right multiplication by a stan- dard generator x0 or x1. We enumerate the average number of carets along the left edge of an n-caret tree, and use an Euler transformation to make some conjectures relating to right multiplication by a generator. We describe a computer algorithm which produces Fordham’s Table, and discuss using the computer algorithm to find a corresponding Fordham’s Table for different generating sets for F. We expound upon the work of Cleary and Taback by completely classifying dead end elements of Thompson’s group, and use the classification to discuss the spread of dead end elements and describe interesting elements we call deep roots. We discuss how deep roots may aid in answering the amenability problem for Thompson’s group. The second part of our work deals with random facet pairings of simplexes. We show that a random endpoint pairings of segments most often results in a disconnected one-manifold, and relate this to a game called “The Human Knot.” When the dimension of the simplexes is greater than 1, however, a random facet pairing most often results in a connected pseudo manifold. This result can be stated in terms of graph theory as follows. Most regular multi graphs are connected, as long as the common valence is at least three. ACKNOWLEDGMENTS I would like to thank my advisor Jim Cannon for the many hours he has spent helping me with this work, as well as for his encouragement and support. In addition I would like to thank Luke Henderson, Sharleen De Gaston, and John Bankhead for the time they spent over the summer of 2003 as this project began. The ideas they brought helped to fuel some of the results contained herein. I will try to make mention of the results which stemmed from their work. Table of Contents I Statistical Properties of Thompson’s Group 1 1 Introduction 4 1.1 Thompson’sGroup ........................... 4 1.2 PresentationsandNormalForms . 15 1.3 Amenability............................... 18 2 Enumerating Thompson’s Group by Number of Carets 26 2.1 The Number of Reduced Pairs of n-caretTrees. 28 2.2 Asymptotics and Probabilities . 36 2.2.1 RandomPairsofCaretTrees . 38 2.2.2 UpperandLowerBounds . 51 2.3 TreeFunctions ............................. 54 2.4 Counting the number of n-caret trees with a prescribed set of exposed carets .................................. 57 2.5 How Multiplication Affects the Number Of Carets . .. 60 3 Fordham’s Table 65 3.1 NotationandFordham’sTheorem . 67 3.2 Discovering Fordham’s Table Algebraically . .... 68 3.3 ANewGeneratingSet ......................... 70 4 Dead End Elements 73 4.1 A Characterization of Dead End Elements . 74 4.2 TheSpreadofDeadEnds ....................... 82 4.3 DeepRoots ............................... 86 II On Random Pseudo Manifolds 89 5 Random One-Manifolds 91 5.1 TheHumanKnot............................ 92 5.2 Modifications .............................. 95 6 Pseudo Manifolds and Brooms 98 6.1 BroomsandNotation. .. .. 100 6.2 Most Pseudo Manifolds are Connected . 101 6.3 AddingStrawstotheBrooms . 106 References 113 viii List of Tables 1 The Number Nn of Elements of Thompson’s Group F ....... 29 2 The Difference Triangle of the Catalan Numbers . 41 3 The Numbers LL(n, k) ......................... 63 4 Difference triangles of LL(n, 2)..................... 64 5 Blake’sTable .............................. 68 6 Effects on word length based on a caret type change. 77 7 The effects of multiplication by x0 onwordlength.. 78 1 8 The effects of multiplication by x0− onwordlength. 79 9 The effects of multiplication by x1 onwordlength.. 79 1 10 The effects of multiplication by x1− onwordlength. 80 11 TheHumanKnotProbabilities . 93 12 The Human Knot Modified Probabilities . 98 ix List of Figures 1 The Infinite Binary Tree T ....................... 5 2 NumberingtheLeaves ......................... 6 3 NumberingtheCarets ......................... 7 4 The infinite binary tree T subdivides [0, 1] ............. 7 5 Partitions of [0, 1] ........................... 8 6 The generator x0. ........................... 9 7 AnEquivalentPairofTrees ...................... 9 8 AStackedTreePair .......................... 10 9 ASimpleReduction .......................... 11 10 AMoreComplexReduction .. .. 11 11 ExposedCarets ............................. 11 12 ReducingUsingCarets ......................... 13 13 MultiplyingTrees............................ 14 14 The generator x1. ........................... 16 15 The Rotational Effects of applying x0 and x1. ............ 17 16 The Cayley Graph of Z ......................... 19 17 The Cayley Graph of Z Z ...................... 20 × 18 The Cayley Graph of Z Z ....................... 21 19 VerticalCayleyGraphs.∗ . .. .. 24 20 LocalDeadEndElementStructure . 25 21 LocalDeepRootStructure. 26 22 CaretTypes............................... 66 23 The element y1. ............................ 71 24 The General Tree Pair Diagram of Dead End Elements . 75 25 TheLocalSpreadofDeadEndElements . 83 26 TheGeneralTreePairDiagramofDeepRoots . 87 27 TheHumanKnotGame ........................ 90 x Part I Statistical Properties of Thompson’s Group In 1965, Thompson’s group F was used to find the first example of a finitely pre- sented infinite simple group. Since then, F has appeared in other branches of mathematics, as well as computer science. In particular F is one of the simplest non-trivial examples of a diagram group (see [9]) and it appears almost anywhere there are group actions on binary trees. Thompson’s group appears in homotopy theory and splitting homotopy idempotents. Other places where Thompson’s group has been used include: proper isometric actions on Hilbert space, symmetries of the modular tower of genus zero real stable curves, the first example of an FP finitely ∞ generated group, and the list goes on. For many years, mathematicians have been trying to solve the amenability problem for Thompson’s group F . The history of amenability dates back to the trying to find an ideal measure. The Hahn-Banach Theorem was invented to prove that Abelian groups are amenable. The Banach- Tarski paradox is directly related to amenability. Thompson’s group is arguably the most famous group for which the amenability problem is still a question. Answering the amenability problem for Thompson’s group should provide new insights into the study of amenability and the geometrical structure of F . Our work is a statistical and geometric study of properties of Thompson’s Group F . Some of these proper- ties may help in understanding the geometry of Thompson’s group as it relates to amenability. In this dissertation we prove the following: 1 The number of elements N of Thompson’s Group F represented by a reduced • n n-caret tree pair is (Theorem 2.8) n/2 k d e n 2k+1 n 1 i k Nn = 2 − − Cn k( 1) Cn i n 2k +1 − − i − i=0 Xk=1 X − where Cn denotes the number of n-caret trees. This gives a type of growth function for
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