STUDENT MATHEMATICAL LIBRARY Volume 81 From Groups to Geometry and Back Vaughn Climenhaga Anatole Katok American Mathematical Society Mathematics Advanced Study Semesters 2010 Mathematics Subject Classification. Primary20-01, 51-01; Secondary 20F65, 22E40, 22F50, 51M05, 51M10, 54H15, 57M10, 57M60. For additional informationand updates on this book, visit www.ams.org/bookpages/stml-81 Library of Congress Cataloging-in-Publication Data Names: Climenhaga,Vaughn,1982-|Katok,A.B. Title: Fromgroupstogeometryandback/VaughnClimenhaga,AnatoleKatok. Description: Providence, Rhode Island: AmericanMathematicalSociety, [2017] |Series: Studentmathematicallibrary;volume81|“MathematicsAdvanced StudySemesters.” |Includesbibliographicalreferencesandindex. Identifiers: LCCN2016043600|ISBN9781470434793(alk. paper) Subjects: LCSH: Group theory. | Number theory. | Topology. | Geometry. | Mathematical analysis. | AMS: Group theory and generalizations – Instruc- tionalexposition(textbooks,tutorialpapers,etc.). msc|Geometry–Instruc- tional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations–Specialaspectsofinfiniteorfinitegroups–Geometricgroup theory. msc | Topological groups, Lie groups – Lie groups – Discrete sub- groups of Lie groups. msc | Topological groups, Lie groups – Noncompact transformationgroups–Groupsasautomorphismsofotherstructures. msc| Geometry–Realandcomplexgeometry–Euclideangeometries(general)and generalizations. msc | Geometry – Real and complex geometry – Hyperbolic andellipticgeometries(general)andgeneralizations. msc|Generaltopology– Connectionswithotherstructures,applications–Transformationgroupsand semigroups. msc | Manifolds and cell complexes – Low-dimensional topology – Covering spaces. msc | Manifolds and cell complexes – Low-dimensional topology–Groupactionsinlowdimensions. msc Classification: LCC QA174.2.C55 2017| DDC512/.2–dc23LC recordavailable athttps://lccn.loc.gov/2016043600 (cid:2)c 2017bytheauthors. Allrightsreserved. PrintedintheUnitedStatesofAmerica. Contents Foreword: MASS at Penn State University xi Preface xiii Guide for instructors xvii Chapter 1. Elements of group theory 1 Lecture 1. First examples of groups 1 a. Binary operations 1 b. Monoids, semigroups, and groups 3 c. Examples from numbers and multiplication tables 8 Lecture 2. More examples and definitions 11 a. Residues 11 b. Groups and arithmetic 13 c. Subgroups 15 d. Homomorphisms and isomorphisms 18 Lecture 3. First attempts at classification 20 a. Bird’s-eye view 20 b. Cyclic groups 22 c. Direct products 25 d. Lagrange’s Theorem 29 Lecture 4. Non-abelian groups and factor groups 31 a. The first non-abelian group and permutation groups 31 b. Representations and group actions 34 c. Automorphisms: Inner and outer 38 d. Cosets and factor groups 42 Lecture 5. Groups of small order 46 a. Structure of finite groups of various orders 46 b. Back to permutation groups 50 c. Parity and the alternating group 54 Lecture 6. Solvable and nilpotent groups 57 a. Commutators: Perfect, simple, and solvable groups 57 b. Solvableandsimplegroupsamongpermutationgroups 60 c. Solvability of groups and algebraic equations 65 d. Nilpotent groups 65 Chapter 2. Symmetry in the Euclidean world: Groups of isometries of planar and spatial objects 69 Lecture 7. Isometries of R2 and R3 69 a. Groups related to geometric objects 69 b. Symmetries of bodies in R2 72 c. Symmetries of bodies in R3 77 Lecture 8. Classifying isometries of R2 82 a. Isometries of the plane 82 b. Even and odd isometries 83 c. Isometries are determined by three points 85 d. Isometries are products of reflections 87 e. Isometries in R3 90 Lecture 9. The isometry group as a semidirect product 91 a. The group structure of Isom(R2) 91 b. Isom+(R2) and its subgroups G+ and T 94 p c. Internal and external semidirect products 96 d. Examples and properties of semidirect products 98 Lecture 10. Discrete isometry groups in R2 100 a. Finite symmetry groups 100 b. Discrete symmetry groups 105 c. Quotient spaces by free and discrete actions 112 Lecture 11. Isometries of R3 with fixed points 114 a. Classifying isometries of R3 114 b. Isometries of the sphere 117 c. The structure of SO(3) 118 d. The structure of O(3) and odd isometries 120 Lecture 12. Finite isometry groups in R3 121 a. Finite rotation groups 121 b. Combinatorial possibilities 126 c. A unique group for each combinatorial type 129 Lecture 13. The rest of the story in R3 133 a. Regular polyhedra 133 b. Completion of classification of isometries of R3 139 Lecture 14. A more algebraic approach 143 a. From synthetic to algebraic: Scalar products 143 b. Convex polytopes 148 c. Regular polytopes 150 Chapter 3. Groups of matrices: Linear algebra and symmetry in various geometries 155 Lecture 15. Euclidean isometries and linear algebra 155 a. Orthogonal matrices and isometries of Rn 155 b. Eigenvalues, eigenvectors, and diagonalizable matrices 158 c. Complexification, complex eigenvectors, and rotations 161 d. Differing multiplicities and Jordan blocks 163 Lecture 16. Complex matrices and linear representations 165 a. Hermitian product and unitary matrices 165 b. Normal matrices 170 c. Symmetric matrices 173 d. Linear representations of isometries and more 174 Lecture 17. Other geometries 176 a. The projective line 176 b. The projective plane 181 c. The Riemann sphere 185 Lecture 18. Affine and projective transformations 187 a. Review of various geometries 187 b. Affine geometry 190 c. Projective geometry 195 Lecture 19. Transformations of the Riemann sphere 197 a. Characterizing fractional linear transformations 197 b. Products of circle inversions 199 c. Conformal transformations 202 d. Real coefficients and hyperbolic geometry 203 Lecture 20. A metric on the hyperbolic plane 204 a. Ideal objects 204 b. Hyperbolic distance 205 c. Isometries of the hyperbolic plane 209 Lecture 21. Solvable and nilpotent linear groups 212 a. Matrix groups 212 b. Upper-triangular and unipotent groups 214 c. The Heisenberg group 216 d. The unipotent group is nilpotent 218 Lecture 22. A little Lie theory 221 a. Matrix exponentials 221 b. Lie algebras 224 c. Lie groups 227 d. Examples 229 Chapter 4. Fundamental group: A different kind of group associated to geometric objects 233 Lecture 23. Homotopies, paths, and π 233 1 a. Isometries vs. homeomorphisms 233 b. Tori and Z2 235 c. Paths and loops 238 d. The fundamental group 241 e. Algebraic topology 246 Lecture 24. Computation of π for some examples 246 1 a. Homotopy equivalence and contractible spaces 246 b. The fundamental group of the circle 251 c. Tori and spheres 253 d. Abelian fundamental groups 255 Lecture 25. Fundamental group of a bouquet of circles 256 a. Covering of bouquets of circles 256 b. Standard paths and elements of the free group 264 Chapter 5. From groups to geometric objects and back 269 Lecture 26. The Cayley graph of a group 269 a. Finitely generated groups 269 b. Finitely presented groups 273 c. Free products 279 Lecture 27. Subgroups of free groups via covering spaces 280 a. Homotopy types of graphs 280 b. Covering maps and spaces 283 c. Deck transformations and group actions 286 d. Subgroups of free groups are free 289 Lecture28. Polygonalcomplexesfromfinitepresentations 290 a. Planar models 290 b. The fundamental group of a polygonal complex 296 Lecture 29. Isometric actions on H2 302 a. Hyperbolic translations and fundamental domains 302 b. Existence of free subgroups 307 Lecture 30. Factor spaces defined by symmetry groups 311 a. Surfaces as factor spaces 311 b. Modular group and modular surface 315 c. Fuchsian groups 320 d. Free subgroups in Fuchsian groups 322 e. The Heisenberg group and nilmanifolds 325 Lecture 31. More about SL(n,Z) 327 a. Generators of SL(2,Z) by algebraic method 327 b. The space of lattices 329 c. The structure of SL(n,Z) 331 d. Generators and generating relations for SL(n,Z) 333 Chapter 6. Groups at large scale 337 Lecture 32. Introduction to large scale properties 337 a. Commensurability 338 b. Growth rates in groups 340 c. Preservation of growth rate 345 Lecture 33. Polynomial and exponential growth 348 a. Dichotomy for linear orbits 348 b. Natural questions 349 c. Growth rates in nilpotent groups 351 d. Milnor–Wolf Theorem 354 Lecture 34. Gromov’s Theorem 356 a. General ideas 356 b. Large scale limit of two examples 359 c. General construction of a limiting space 364 Lecture 35. Grigorchuk’s group of intermediate growth 366 a. Automorphisms of binary trees 366 b. Superpolynomial growth 369 c. Subexponential growth 372 Lecture 36. Coarse geometry and quasi-isometries 376 a. Coarse geometry 376 b. Groups as geometric objects 380 c. Finitely presented groups 382 Lecture 37. Amenable and hyperbolic groups 385 a. Amenability 385 b. Conditions for amenability and non-amenability 387 c. Hyperbolic spaces 390 d. Hyperbolic groups 393 e. The Gromov boundary 393 Hints to selected exercises 395 Suggestions for projects and further reading 401 Bibliography 409 Index 413 Foreword: MASS at Penn State University This book is part of a collection published jointly by the American Mathematical Society and the MASS (Mathematics AdvancedStudy Semesters) program as a part of the Student Mathematical Library series. The books in the collection are based on lecture notes for advanced undergraduate topics courses taught at the MASS (Math- ematics Advanced Study Semesters) program at Penn State. Each book presents a self-contained exposition of a non-standard mathe- matical topic, often related to current research areas, which is acces- sible to undergraduate students familiar with an equivalent of two years of standard college mathematics, and is suitable as a text for an upper division undergraduate course. Started in 1996, MASS is a semester-long program for advanced undergraduatestudentsfromacrosstheUSA.Theprogram’scurricu- lum amounts to sixteen credit hours. It includes three core courses fromthegeneralareasofalgebra/numbertheory,geometry/topology, and analysis/dynamical systems, custom designed every year; an in- terdisciplinary seminar; and a special colloquium. In addition, ev- ery participant completes three research projects, one for each core course. The participants are fully immersed into mathematics, and this,aswellasintensiveinteractionamongthestudents,usuallyleads toadramaticincreaseintheirmathematicalenthusiasmandachieve- ment. The program is unique for its kind in the United States. DetailedinformationabouttheMASSprogramatPennStatecan be found on the website www.math.psu.edu/mass.