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Isometries and CAT(O) Metric Spaces Isometries and CAT(O) Metric Spaces by Naomi Lynne Wolfson, B.Math A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science School of Mathematics and Statistics Ottawa-Carleton Institute for Mathematics and Statistics Carleton University Ottawa, Ontario, Canada April 27, 2006 © copyright 2006, Naomi Lynne Wolfson Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Library and Bibliotheque et Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-16506-5 Our file Notre reference ISBN: 978-0-494-16506-5 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce,Canada de reproduire, publier, archiver, publish, archive, preserve, conserve,sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet,distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform,et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these. While these forms may be includedBien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. i * i Canada Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract Using the CAT(O) inequality, non-positively curvature in the Riemanian sense can be generalized to the much broader setting of geodesic metric spaces. By studying CAT(O) spaces and groups which act on them properly and cocom- pactly by isometries, a great deal can be learned about the structure of both. This thesis presents an introduction to CAT(O) spaces and explores the conse­ quences of proper cocompact group actions on these spaces, including the Flat Torus Theorem, the Splitting Theorem, and a solution to the word problem for such groups. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ii Acknowledgments I would like to thank my supervisor Dr. Ben Steinberg as well as Dr. Inna Bumagin, Dr. Mike Moore and Valerie Daley from the School of Mathematics and Statistics at Carleton University. Above all else, I would like to thank my parents Eleanor Bennett and Michael Wolfson, who have always supported me throughout all my crazy endeavors and without whom I would never have gotten this far. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents A b stract....................................................................................................... Acknowledgments ....................................................................................... ii List of Figures.............................................................................................. iv 0 Introduction 1 1 Geodesics and the Model Spaces 4 1.1 What are Geodesics? ...................................................................... 4 1.2 Geodesic Metric Spaces and Convexity ...................................... 9 1.3 Geodesic Triangles and Angles Between Geodesics ..................... 14 1.4 The Euclidean, Spherical, and Hyperbolic Metric Spaces . 22 1.5 The Model S p a c e s ................................................................... 27 2 Introduction to CAT(k) Spaces 36 2.1 The CAT( k) Inequality .................................................................... 36 2.2 Insights into Convexity .................................................................... 44 2.3 Projection onto a Convex Subspace ............................................. 50 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONTENTS iv 2.4 The Centre of a Bounded Set ....................................................... 55 2.5 Flat Subspaces and the Product Decomposition Theorem . 57 3 Group Actions 75 3.1 Basics of Group A c tio n s ................................................................ 75 3.2 Group Presentations and A ctions ............................................... 83 3.3 Bruhat-Tits Fixed Point Theorem ............................................... 93 4 Decision Problems 96 4.1 The Word Problem ......................................................................... 96 5 Investigations of Isometries 110 5.1 Displacement Functions and Translation Length ...................... 110 5.2 Three Classes of Isometries............................................................. 117 6 Groups of Isometries 127 6.1 More on the Structure of Isometries ............................................. 127 6.2 Splitting CAT(O) Spaces................................................................ 132 6.3 The Flat Torus Theorem ................................................................ 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 1.1 Section of the Cayley graph of F(a, b) .......................................... 13 1.2 Geodesic Triangle ............................................................................ 14 1.3 Comparison Triangle ...................................................................... 15 1.4 Comparison Triangle ...................................................................... 17 1.5 Split a into the sum of a' and a " ............................................... 20 1.6 Split the geodesic segment [a, 6] by a and a ' .............................. 21 1.7 Euclidean Law of Cosines ............................................................ 23 1.8 Spherical Distances ......................................................................... 24 1.9 Spherical Law of Cosines ............................................................... 25 1.10 Hyperbolic Law of Cosines ............................................................ 26 1.11 ^-comparison Triangle ................................................................... 30 1.12 Quadrilateral with vertices A, B, B and C .............................. 32 1.13 Construction of B' ......................................................................... 33 1.14 Construction of Triangle A(H, B, B ' ) .......................................... 35 2.1 Comparison Point ............................................................................. 37 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES vi 2.2 Comparison Triangle for A (\p,q]',\p,x\,[x,q]) 40 2.3 Comparison Triangle For c(0), c(t0), c(t0 + e ) ............................... 42 2.4 Graph of a Convex F u n ctio n .......................................................... 45 2.5 Comparison Triangle for (c(0), c(l), c '( l ) ) ..................................... 47 2.6 Geodesic paths c, d and c " .............................................................. 48 2.7 Comparison Points Relative to the Annular R egion ............... 51 2.8 Location of p e [tt(x ),I7]............................................................... 53 2.9 Arrangement of A' and A" in E 2 ............................................... 59 2.10 Location of x and x' in A(p,q,q') ............................................... 60 2.11 Placement of Ai and A 2 ................................................................ 63 2.12 Calculating the distance between cXl(ti) and cX2(£2) ............... 73 3.1 Constructing the path cs ................................................................ 86 3.2 Relationship between D, CU3(r), and X ................................... 89 3.3 Selection of 5i and 5 [ ...................................................................... 90 4.1 Dividing the geodesic segment [x 0,7 • xo] into f°ur parts .... 99 4.2 A(xo, 7 • xo, 7 ' • Xo) with a and b illustrated ................................ 103 4.3 Comparing isosceles triangles ...................................................... 104 4.4 7 and 7 ' in the metric Cayley graph of T ..................................... 106 5.1 Constructing a quadrilateral in E 2 for com parison ................... 115 6.1 Commutative Diagram for 72 , p, and p1 ...................................... 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 0 Introduction The
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