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  • Quasiconvexity in Word-Hyperbolic Coxeter Groups

    Adam Simon LevineMathematics Department

    Harvard University

    April 3, 2005

  • Abstract

    In Chapter 1, I outline many of the basic properties of word-hyperbolic groups and givean introduction to CAT() spaces. In Chapter 2, I prove a number of important theoremsabout Coxeter groups, including the Exchange Condition and Titss solution to the wordproblem. I also discuss Moussongs theorem on which Coxeter groups are word-hyperbolic.Finally, in Chapter 3, I present some original research showing that certain subgroups ofcertain word-hyperbolic Coxeter groups are quasiconvex.

  • Acknowledgements

    I am deeply grateful to the following people for their assistance with this thesis:To my advisor, Prof. Dylan Thurston, for guiding me through the thesis-writing process,

    always taking the time to answer my questions, and constantly challenging me to learn more.To Profs. Patrick Bahls and Kim Whittlesey, for supervising my summer research at the

    University of Illinois at Urbana-Champaign and sparking my interest in geometric grouptheory.

    To Mark Lipson, for his meticulous proofreading and his pages of detailed suggestionsand corrections.

    To Peter Green and David Escott, for their helpful comments.To my roommates, for putting up with my math puns for four years.To Amy Pasternack, for her late-night cookie deliveries to the Science Center computer

    lab.To my brother Ben, for being a good sport during my lengthy explanations of Coxeter

    groups, and for being a dear friend.To my parents, for their helpful advice on how to make my thesis comprehensible to

    laypeople, their valiant efforts to explain to friends and relatives exactly what I study, andtheir constant love and support.

    And finally, to my high school calculus teacher, Mr. Robert Arrigo, for instilling in me alifelong love of mathematics.

  • Contents

    List of Figures 2

    Introduction 3

    1 The Theory of Word-Hyperbolic Groups 111.1 -Hyperbolic Metric Spaces and Quasi-Isometries . . . . . . . . . . . . . . . 111.2 Word-Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 The Rips Complex and Finite Presentations . . . . . . . . . . . . . . . . . . 181.4 Quasiconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5 Spaces of Non-Positive Curvature . . . . . . . . . . . . . . . . . . . . . . . . 25

    2 Coxeter Groups 282.1 The Geometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 The Exchange and Deletion Conditions . . . . . . . . . . . . . . . . . . . . . 322.3 Discrete Reflection Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 The Word Problem in Coxeter Groups . . . . . . . . . . . . . . . . . . . . . 412.5 Parabolic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.6 Word-Hyperbolic Coxeter Groups . . . . . . . . . . . . . . . . . . . . . . . . 45

    3 Original Results 50

    Poetic Conclusion 56

    Bibliography 58

    1

  • List of Figures

    1 Reflections through lines meeting at 60. . . . . . . . . . . . . . . . . . . . . 42 Cayley graph for Coxeter group generated by a, b, and c, subject to the rules

    aba = bab, bcb = cbc, and ac = ca. . . . . . . . . . . . . . . . . . . . . . . . 53 Cayley graph for Coxeter group generated by a, b, and c, subject to the rules

    aba = bab, bcb = cbc, and aca = cac. . . . . . . . . . . . . . . . . . . . . . . 64 The hyperboloid (left) and Poincare (right) models of the hyperbolic plane. 75 Cayley graph for the (2, 4, 6) triangle group, embedded in the Poincare model

    of the hyperbolic plane H2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 M.C. Eschers woodcut Circular Limit I [1, p. 319]. . . . . . . . . . . . . . 10

    7 A -slim triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Schematic of the proof of Theorem 1.4. . . . . . . . . . . . . . . . . . . . . 149 The two cases in the proof of Lemma 1.13. . . . . . . . . . . . . . . . . . . . 2010 A k-quasiconvex subspace Y . . . . . . . . . . . . . . . . . . . . . . . . . . . 2211 The Cayley graphs CA(Z2) (left) and CB(Z2) (right) for Example 1.16. . . . 2312 Schematic of the proof of Theorem 1.18. . . . . . . . . . . . . . . . . . . . . 24

    13 The positions of the roots v.es in the proof of Lemma 2.5 when ms,t = 5, alongwith the fundamental domain K defined at the end of the section. . . . . . 34

    14 Coxeter diagrams for the irreducible finite and affine Coxeter groups. . . . . 4015 The Coxeter cell C(W ) and block B(W ) for the (2, 2, 3) triangle group. . . 47

    16 Schematic of the proof of Theorem 3.2. . . . . . . . . . . . . . . . . . . . . . 5317 The two types of cancellation diagrams in Proposition 3.4. . . . . . . . . . . 55

    2

  • Introduction

    Given a finite set of letters, consider the set of words not English words, but simply finitesequences of letters that can be written with those letters. We define two operations:

    Whenever a letter appears twice consecutively, we may delete both occurrences. Like-wise, we may insert two consecutive, identical letters into a word at any point.

    For each pair of letters (say, a and b), choose one of the following rules: ab = ba,aba = bab, abab = baba, and so on. That is, if we see a sequence of the form aba ,we may replace it with bab (where both sequences are of the specified length), andvice versa. These rules need not be the same for different pairs of letters; we couldsay, for instance, that ab = ba, aca = cac, and bcbc = cbcb. We may also choose not toimpose any such rule on particular pairs of letters.

    Two words are said to be equivalent if it is possible to get from one to the other using thesetwo operations. The set of all classes of equivalent words is called a Coxeter group (namedfor the geometer H.S.M. Coxeter, who studied this construction starting in the 1930s). Forinstance, if the rules are ab = ba, aca = cac, and bcbc = cbcb as above, then the wordsabac, aabc, baac, and bc are all equivalent, so they represent the same element of the Coxetergroup. They are not equivalent, however, to the word cb, which thus represents a differentelement of the Coxeter group.1

    As an example, let the set of letters (also called generators) consist of a and b, andsuppose that aba = bab. The elements of the Coxeter group can be written as a, b, ab, ba,aba (which is the same as bab), and the so-called empty word, a word of zero letters, oftendenoted by 1. Any longer word can be reduced to one of these six possibilities. As anotherexample, the group generated by a, b, and c, with the rules aba = bab, bcb = cbc, andac = ca, has 24 distinct elements. On the other hand, if the rules are aba = bab, bcb = cbc,and aca = cac, then the group contains infinitely many distinct elements, for the words abc,abcabc, abcabcabc, abcabcabcabc, etc. all represent different elements.

    Many Coxeter groups occur naturally as so-called reflection groups, sets of transforma-tions of a space that can be realized as the composition of a given set of reflections. Forinstance, consider two lines in the plane, labelled La and Lb, that intersect at an angle of 60

    .

    1Formally, in order to call this set a group, we need to define a law of composition. However, as thisIntroduction is intended for a general audience, we shall use this colloquial description. A rigorous definitionof Coxeter groups will appear in Chapter 2.

    3

  • La

    Lb

    Figure 1: Reflections through lines meeting at 60.

    Take any figure, such as the upper right letter R shown in Figure 1, and reflect it across theline La to obtain the upside-down R in the lower-right corner. Next, reflect the new figureacross Lb to obtain the R in the upper-left corner, and then reflect that down across La tothe lower-left backward R. Next, starting with the original figure again, reflect first acrossLb, then across La, then across Lb again, and note that the final image is exactly the same.In other words, if a and b represent reflections across La and Lb, respectively, then we obtainthe rule aba = bab. Also, performing the same reflection twice in a row leaves every pointfixed, so we may insert or delete pairs of consecutive, identical reflections into any sequencewithout consequence. Thus, we can say that the set of transformations of the plane thatare compositions of the reflections a and b is a Coxeter group. More generally, if La and Lbintersect instead at an angle of 180/n, where n is an integer greater than 1, we obtain aCoxeter group with the rule

    aba n letters

    = bab n letters

    .

    One way to visualize a Coxeter group geometrically is the Cayley graph. This is a graph(a network of vertices connected by edges) with one vertex for every distinct element ofthe group and with edges labelled with the different letters, one of each type at each ver-tex. Words written with the generating letters correspond to paths emanating from a fixedstarting point, and two words represent the same element of the group if and only if thecorresponding paths end at the same point.

    For example, the Cayley graph for the Coxeter group generated by a and b with the rulethat ab = ba is a square with its sides alternately labelled a and b. If aba = bab, the graphis a hexagon; if abab = baba, an octagon; and so on. Assume that the polygons are regular;that is, all the sides and all the angles are equal.2 With more than two letters, the graph

    2Formally, the angle measures are a property not of the graph itself but of the embedding of the graphinto a larger space, in this case Rn. To avoid having to introduce the concept of an abstractly defined graph,

    4

  • Figure 2: Cayley graph for Coxete

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