از گروه‌ها تا هندسه و برعکس (کتابخانه ریاضی دانشجویی) (کتابخانه ریاضی دانشجویی، ۸۱)

Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering spaces and deck transformations. In the other direction, Cayley graphs, planar models, and fundamental domains appear as geometric objects associated to groups. The final chapter discusses groups themselves as geometric objects, including a gentle introduction to Gromov's theorem on polynomial growth and Grigorchuk's example of intermediate growth. The book is accessible to undergraduate students (and anyone else) with a background in calculus, linear algebra, and basic real analysis, including topological notions of convergence and connectedness. This book is a result of the MASS course in algebra at Penn State University in the fall semester of 2009. This book is published in cooperation with Mathematics Advanced Study Semesters. Cover 1 Title page 4 Contents 6 Foreword: MASS at Penn State University 12 Preface 14 Guide for instructors 18 Chapter 1. Elements of group theory 22 Lecture 1. First examples of groups 22 a. Binary operations 22 b. Monoids, semigroups, and groups 24 c. Examples from numbers and multiplication tables 29 Lecture 2. More examples and definitions 32 a. Residues 32 b. Groups and arithmetic 34 c. Subgroups 36 d. Homomorphisms and isomorphisms 39 Lecture 3. First attempts at classification 41 a. Bird’s-eye view 41 b. Cyclic groups 43 c. Direct products 46 d. Lagrange’s Theorem 50 Lecture 4. Non-abelian groups and factor groups 52 a. The first non-abelian group and permutation groups 52 b. Representations and group actions 55 c. Automorphisms: Inner and outer 59 d. Cosets and factor groups 63 Lecture 5. Groups of small order 67 a. Structure of finite groups of various orders 67 b. Back to permutation groups 71 c. Parity and the alternating group 75 Lecture 6. Solvable and nilpotent groups 78 a. Commutators: Perfect, simple, and solvable groups 78 b. Solvable and simple groups among permutation groups 81 c. Solvability of groups and algebraic equations 86 d. Nilpotent groups 86 Chapter 2. Symmetry in the Euclidean world: Groups of isometries of planar and spatial objects 90 Lecture 7. Isometries of \RR2 and \RR3 90 a. Groups related to geometric objects 90 b. Symmetries of bodies in \RR2 93 c. Symmetries of bodies in \RR3 98 Lecture 8. Classifying isometries of \RR2 103 a. Isometries of the plane 103 b. Even and odd isometries 104 c. Isometries are determined by three points 106 d. Isometries are products of reflections 108 e. Isometries in \RR3 111 Lecture 9. The isometry group as a semidirect product 112 a. The group structure of \Isom(\RR2) 112 b. \Isom+(\RR2) and its subgroups G_{\pp}+ and \TTT 115 c. Internal and external semidirect products 117 d. Examples and properties of semidirect products 119 Lecture 10. Discrete isometry groups in \RR2 121 a. Finite symmetry groups 121 b. Discrete symmetry groups 126 c. Quotient spaces by free and discrete actions 133 Lecture 11. Isometries of \RR3 with fixed points 135 a. Classifying isometries of \RR3 135 b. Isometries of the sphere 138 c. The structure of SO(3) 139 d. The structure of O(3) and odd isometries 141 Lecture 12. Finite isometry groups in \RR3 142 a. Finite rotation groups 142 b. Combinatorial possibilities 147 c. A unique group for each combinatorial type 150 Lecture 13. The rest of the story in \RR3 154 a. Regular polyhedra 154 b. Completion of classification of isometries of \RR3 160 Lecture 14. A more algebraic approach 164 a. From synthetic to algebraic: Scalar products 164 b. Convex polytopes 169 c. Regular polytopes 171 Chapter 3. Groups of matrices: Linear algebra and symmetry in various geometries 176 Lecture 15. Euclidean isometries and linear algebra 176 a. Orthogonal matrices and isometries of \RRn 176 b. Eigenvalues, eigenvectors, and diagonalizable matrices 179 c. Complexification, complex eigenvectors, and rotations 182 d. Differing multiplicities and Jordan blocks 184 Lecture 16. Complex matrices and linear representations 186 a. Hermitian product and unitary matrices 186 b. Normal matrices 191 c. Symmetric matrices 194 d. Linear representations of isometries and more 195 Lecture 17. Other geometries 197 a. The projective line 197 b. The projective plane 202 c. The Riemann sphere 206 Lecture 18. Affine and projective transformations 208 a. Review of various geometries 208 b. Affine geometry 211 c. Projective geometry 216 Lecture 19. Transformations of the Riemann sphere 218 a. Characterizing fractional linear transformations 218 b. Products of circle inversions 220 c. Conformal transformations 223 d. Real coefficients and hyperbolic geometry 224 Lecture 20. A metric on the hyperbolic plane 225 a. Ideal objects 225 b. Hyperbolic distance 226 c. Isometries of the hyperbolic plane 230 Lecture 21. Solvable and nilpotent linear groups 233 a. Matrix groups 233 b. Upper-triangular and unipotent groups 235 c. The Heisenberg group 237 d. The unipotent group is nilpotent 239 Lecture 22. A little Lie theory 242 a. Matrix exponentials 242 b. Lie algebras 245 c. Lie groups 248 d. Examples 250 Chapter 4. Fundamental group: A different kind of group associated to geometric objects 254 Lecture 23. Homotopies, paths, and π1 254 a. Isometries vs. homeomorphisms 254 b. Tori and \ZZ2 256 c. Paths and loops 259 d. The fundamental group 262 e. Algebraic topology 267 Lecture 24. Computation of π1 for some examples 267 a. Homotopy equivalence and contractible spaces 267 b. The fundamental group of the circle 272 c. Tori and spheres 274 d. Abelian fundamental groups 276 Lecture 25. Fundamental group of a bouquet of circles 277 a. Covering of bouquets of circles 277 b. Standard paths and elements of the free group 285 Chapter 5. From groups to geometric objects and back 290 Lecture 26. The Cayley graph of a group 290 a. Finitely generated groups 290 b. Finitely presented groups 294 c. Free products 300 Lecture 27. Subgroups of free groups via covering spaces 301 a. Homotopy types of graphs 301 b. Covering maps and spaces 304 c. Deck transformations and group actions 307 d. Subgroups of free groups are free 310 Lecture 28. Polygonal complexes from finite presentations 311 a. Planar models 311 b. The fundamental group of a polygonal complex 317 Lecture 29. Isometric actions on \HH2 323 a. Hyperbolic translations and fundamental domains 323 b. Existence of free subgroups 328 Lecture 30. Factor spaces defined by symmetry groups 332 a. Surfaces as factor spaces 332 b. Modular group and modular surface 336 c. Fuchsian groups 341 d. Free subgroups in Fuchsian groups 343 e. The Heisenberg group and nilmanifolds 346 Lecture 31. More about SL(n,\ZZ) 348 a. Generators of SL(2,\ZZ) by algebraic method 348 b. The space of lattices 350 c. The structure of SL(n,\ZZ) 352 d. Generators and generating relations for SL(n,\ZZ) 354 Chapter 6. Groups at large scale 358 Lecture 32. Introduction to large scale properties 358 a. Commensurability 359 b. Growth rates in groups 361 c. Preservation of growth rate 366 Lecture 33. Polynomial and exponential growth 369 a. Dichotomy for linear orbits 369 b. Natural questions 370 c. Growth rates in nilpotent groups 372 d. Milnor–Wolf Theorem 375 Lecture 34. Gromov’s Theorem 377 a. General ideas 377 b. Large scale limit of two examples 380 c. General construction of a limiting space 385 Lecture 35. Grigorchuk’s group of intermediate growth 387 a. Automorphisms of binary trees 387 b. Superpolynomial growth 390 c. Subexponential growth 393 Lecture 36. Coarse geometry and quasi-isometries 397 a. Coarse geometry 397 b. Groups as geometric objects 401 c. Finitely presented groups 403 Lecture 37. Amenable and hyperbolic groups 406 a. Amenability 406 b. Conditions for amenability and non-amenability 408 c. Hyperbolic spaces 411 d. Hyperbolic groups 414 e. The Gromov boundary 414 Hints to selected exercises 416 Suggestions for projects and further reading 422 Bibliography 430 Index 434 Back Cover 442