Let f, g be elements of M, the group of Möbius transformations of the extended complex plane Ĉ = C U ∞. We identify each element of M with a 2 × 2 complex matrix with determinant 1. The three complex numbers, β(f) = tr2

(f) - 4,β(g) = tr2

(g) - 4,γ(f,g) = tr[f,g] - 2, define the group ‹f,g› uniquely up to conjugacy whenever γ(f,g) ≠ 0; where tr(f) and tr(g) denote the traces of representive matrices of f and g respectively, [f,g] denotes the multiplicative commutator fgf-1

g-1

. We call these three complex numbers the parameters of ‹f,g›. This thesis is concerned with the parameters of discrete and elementary subgroups of M

Zhang, Qingxiang

English

Massey Research Online

Copyright is owned by the Author of the thesis.  Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only.  The thesis may not be reproduced elsewhere without the permission of the Author.  Parameters of the Two Generator Discrete Elementary Groups A thesis presented in partial fulfillment of the requirements for the degree of :t\Iast er of Science 1n Mathe1natics At Massey University, Albany, New Zealand Qingxiang Zhang 2006 Abstract Let f , g be elements of M , the group of l\kibius transformations of the extended complex plane C = C U oo. We identify each element of M with a 2 x 2 complex matrix with determinant 1. The three complex numbers. /3(!) = tr2 (f) - 4, 13(g) = tr2 (g) - 4. 1 (1. g) = tr[! , g] - 2, define the group (!, g) uniquely up to conjugacy whenever 1 (!, g) i- 0: where tr(! ) and tr(g) denote the trace· of representive mat rices off and g respect ively, [!, g] denotes the multiplicat ive commutator J gJ- 19- 1. We call these three complex numbers the parameters of(!. g). This thesis is concerned with the parameters of d iscrete and element ary subgro ups of M. II Acknowledgments I wis h t o express my gratitude to my supervisor Distinguished Professor Gaven .\Iartin , for his s upport throughout this tlwsis. And also special thanks to him in patiently correcting my English. I owe much to P rofessor David G auld of Auckland Uniwrsity and m~- friend Haydu Cooper of .\Ia ·scy University for patiently proofreading this thesis. I \\·ould like to thank .\ Iassey University fo r gi\·ing me the opportun ity to learn a nd for the Schola rship I have received. I am grateful to the Marsden Fund I have received for this thesis. I would a lso like to thank my hcloved cat :d illie. who was born on 23rd of Februa ry 2006 and sadly killed 011 20th of .\la rch 2007. for her lovely c-ompa11y th roughout this thesis. Last but not least I \YOtild like thank my hus band Bria11 Pool. for his u11co11ditio11a l support and emotional encouragement . Contents Introduction 1 Spherical and Hyperbolic geometries 1.1 Spherical geometry . 1.2 Hyperbolic geometry ..... 1 5 5 7 2 Mobius transformations on lR" 12 2.1 The :.\Iobius group 0 11 i 11 • . • . • • • • 12 2.2 Poincarc extC'nsion . . . . . . . . . . . . 13 2.3 The Discrete C'l<?nwntary subgroups of M 15 3 Triangle groups 1 7 3.1 Triangle' groups in three differ('llt gconH:trics 17 3.2 Euclidean Triangle' groups 18 3.3 Spherical Triangle groups . . . . . . . . . . . 20 4 The parameters of two-generator subgroups of M 24 ,.U Determining 3(!) algebraically . . . . . . . . . . . . . . . . . . 2"1 4.2 The parameters of A 1• 8 1• A5 and their two generator subgroups 25 4.2.l The elements in A.., and their conjugacy classes 25 4.2.2 The elements in S.., and their conjugacy classes . . . . 25 4.2.3 The elements in A5 and their conjugacy classes 26 4.3 Parameters of two generator groups with parabolic elements 27 4.4 T he parameters of two generator discrete elementary groups containing only loxodromic elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 The isolation of A 4 , S4 , A5 31 5.1 previous resu lts . . . . . . . . . . . . . . . . . . . 31 5.2 Isolationof (-1,-2,b1),(-2,-3,b2),(-1,-3,b3). 33 6 Remarks and furthe r research 40 Reference 41 iii Introduction :\Iobius transformations were studied by the German ma thematician A. F . :\Iobius in the 19th century. F . h:tein proved the group of :\Iobius transformations acting on Euclidean n-space is isomorphic to the group of isometrics of hyperbolic (n + 1)-space (sec [17] . page 147). This discovery leads to a deeper understanding of hyperbolic space and relations between conformal geometry of spheres. the models of hyperbolic space they bound and n-dimcnsio n geometrv. Relevant references can be found in the works of Beardon [l]. R atcliffe [17]. Thurston [20] . Gehring and :\Iartin (see for exa mple [5]. [6]. [9] . [10]) and references therein. In recent years. the s t udy of the 3-dimensional hyperbolic orbi folds. which can be represented as // 3/ C "·here H 3 is hyp0rbolic 3-space (d iscussed in Chapter 1) and C is a discrete non-elementary orientation preserving subgroup of the group of thC' isomct r_\· group. has attrnctC'd much attC'ntion. \\·e are concerned here \Yi t h such discrete subgroups C. \\"e shall assume a basic- knowledge of group theory in our discussion. Let M denote thC' group of :\Iobius transforma tions of the form: J(z) = az + b c::. +d \,·hich we associate wit h the matrix A=(:~ ) a. b. c. d E Cad - be = l. a. b. c. cl E Cad - be= l. (1) (2) There are two basic types of discrete subgroup of M: elementary and non-elementary. whose definitions are given in Chapter 2. The discrete non-elementary groups are known as Kleinian groups in memory of the Mathematician F. Klein. All the discrete elementary groups are known and classified (see [1]) , hence the study of Kleinian groups are of interest. But the discreteness or otherwise of a Kleinian group is not easy to establish. Klimenko and Kopteva gave a criterion for discreteness of Kleinian groups with an invariant plane (see [3]) . While for the Kleinian groups without invariant plane, we have only necessary or only sufficient conditions for t he discreteness of such groups. Theorem 5.4.2 of [1] s tates that a non-elementa ry subgroup G of Mis discrete if and only if for each f and g in G, (!, g) is discrete. Thus t he problem of deciding the discreteness or otherwise of G boils down to consideration of the two generator subgroups. We shall study the discreteness of two generator groups (!, g). The advantage of studying a two generator group is that for every such group (!, g), there are three complex numbers corresponding to it, and the necessary or sufficient condition(s) for non-elementary (!, g) to be discrete can 1 Introduction 2 sometimes be described in terms of the e numbers. These three complex numbers a re ;J(f) = tr2 (J) - 4. ,3(g) = tr2 (g) - 4. 1 (!, g) = tr[!, g] - 2, where tr(!) and lr(g) denote the traces of representive matrices of J and g respectively. and [J,g] denotes the multiplicative commutator fgJ- 19- 1• see [7]. These three numbers arc called the parameters of the two-generator group (J, g) and we write par((J.g)) = (,(J.g).J(J).J(g)). These parameters arc independent of the choice of represcnt ive matrices for J and g and define (!. g) uniquely up to conjugacy whenewr , (J. g) -=I= 0. See [7]. Two subgroups Co and C1 of Care coujugate if for some h in C. C0 = hC1h - 1. Conjugate subgroups are the same from a geometric point of vie,,·. For example. if there exists a unique point fixed by all 9o E Co, then there exists a unique point fixed by all g1 E C 1. The volumes of !{3/ G1 and 113 / Go arc the same and so fort h. The study of th<' discreteness of t\,·o generator groups has a rich history. sec all of our references except [15] and [16] . For exa111ple in [l]. Beardon studies necessary condit ions for a two generator Kleinian group by consiclE-ring the displaccmC'nt function . 1 .:: 1------t smh 2p(:::.g:::). Gehring and :-- Iartin obtain cond itions for (!. g) to be discret<' by examining the distances of f. g from t he identity element in M in [6]. Th<'_,· a lso obtain some sharp estimates for the distance between the axes of ellipt ic- elements in a discrete group in [12]. Klimenko and Koptcva found criteria for discreteness of t,,·o generator I--:lei11ia11 groups generated by a hyperbolic clement and an elliptic element of even order ,,·ith intersection axes in [3]. Tlw most well-known necessary theorem in t he subject is due to .Jorgensen (see [l]): Theorem 0 .1. (Jorgensen·s inequality) Suppose that the Mobius transforrnations J and g generate a discrete non-elementary group with 1 (!, g) = "f and J(f) = 13 . then 1--YI + I.Bl ~ 1. (3) This inequality was studied by Troels J0rgensen in [19]. He proved the inequality by the iteration of the rela tion B0 =B, where A and B are the matrices representing J and g respectively. Another inequality was studied by Delin Tan in [2]: Theorem 0.2. Suppose that the Mobius transformations f and g generate a discrete group with --y(J , g) = --y and (3(!) = {J . lf --y-=/= -1, then b+ 11 + lf3+21 ~ 1. (4) Introduction 3 If , = -1 and r3 -/= -2, then Tan used Lemma 2 in his paper to prove (4); This Lemma was proved by the iteration of the relat ion Bo =B, which is essentially the same as J 0gensen 's iteration scheme. Gehring and l\,Iartin proved (3) and ( 4) independently by investigating the two fixed points O and J + 1 of the polynomial trace 1 (!, gJg - 1 ) = ,(, - J) in [-l]. The inequalities (3) and (4) and Gehring and '.\Iartin·s approach to them give a different perspective to look at the conditions for discreteness of (!, g) . The fact is tha t in the space of two generator discrete groups, all two generator Kleinian groups form a closed set. This has essentia lly been proved by J 0rgensen in [19]. We claim that all the element ary groups are isolated from the set of Kleinian groups in this spac . This claim and precise bounds to describe this isola tion in terms of geomet ric quantities as well as the complex parameters are investigated in this and future research. As we kno,v that every two generator group (!. g) has three complex numbers as its parameter . we can therefore view (!. g) as a point in C 3 . the three dimensional complex space. Let D3 be the subset of C 3 which cont a ins all t he param eters of two generator discrete groups. \:Ye prove that whenever (a, b. b' ) E D 3 corresponds to a discrete element ary group. it is isola ted from the point · corresponding to Kleinian groups. We establish the isola tion of (a. b. b' ) by proving an inequa lity of the fo rm I, + al+ Id+ bl 2: c (5) where c is a real positive number and (, . ,8. /3') are the parameters for any Kleinian group. The rea on that the isolat ion of (a . b. b') only depend on a. b will be explained in C hapter 5. but no te here that (5) a lso implies immedia tely tha t I, + a I + I ,6' + b' I 2: c by interch anging the order of the generators. ote a lso that the inequality (5) also indicates a necessary condition for a Kleinian (!, g) to be discrete. This is the main reason for looking at the isolation of discrete elementary groups. The m ain concern of the first part of this thesis is to determine all the possible parameters for discrete elementary groups. These are the points in C 3 that we shall show to be isolated. We then go on to give estimates on this isolation using some of the ideas discussed above (iteration). This recovers some known results and also generates some new ones. Baribeau and Ransford have given a general description of these parameters in [18] . Gehring and Martin have discussed some of them in many of their papers , see for example [4], 5], [13],or [14]. We consider all the parameters for all the discrete elementary groups more specified in this thesis . To this end we start with some preliminary topics such as the spherical and hyperbolic geometries , Mobius transformations , Triangle groups through Chapter 1 to Chapter 3. The results in these three Chapters are a matter of rewriting known facts. Our Introduction 4 main results are stated in Chapter -1. In this Chapter we investigate the parameters under question using a combination of t,,·o methods: geomet ric and algebraic methods. We omit the lengthy but purely elementary computation process and state our final results in three tables. As we ,vill sec in these tables . for some elementary groups. we are able to find the exact parameters: while for others we can only give a general description similar to the results in [18] as there are parametrised families of these groups. For tho e ,,·hose exact parameters are known. we shall investigate their isolation from Klcinian groups by using the inequality of the form (5) . For instance in Chapter 5. we consider several examples to shmv how to derive such inequalities for (-l.-2.b1). (-2. -3.b2). (-l.-3.b3). 

Parameters of the two generator discrete elementary groups : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand

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Parameters of the two generator discrete elementary groups : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand

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