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Lectures on Groups and Their Connections to Geometry Anatole Katok Vaughn Climenhaga Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: katok [email protected] Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: [email protected] Contents Chapter 1. Elements of group theory 1 Lecture 0 Wednesday, Thursday, and Friday, August 26–28 1 a. Binary operations 1 b. Monoids, semigroups, and groups 2 c. Multiplication tables and basic properties 5 d. Groups and arithmetic 7 e. Subgroups 9 f. Homomorphisms and isomorphisms 11 Lecture 1 Monday, August 31 13 a. Generalities 13 b. Cyclic groups 14 c. Direct products 16 d. Classification 17 e. Permutation groups 19 Lecture 2 Wednesday, September 2 21 a. Representations 21 b. Automorphisms: Inner and outer 23 c. Cosets and factor groups 25 d. Permutation groups 27 Lecture 3 Friday, September 4 30 a. Parity and the alternating group 30 b. Solvable groups 31 c. Nilpotent groups 34 Chapter 2. Symmetry in the Euclidean world: Groups of isometries of planar and spatial objects 37 Lecture 4 Wednesday, September 9 37 a. Groups of symmetries 37 b. Symmetries of bodies in R2 39 c. Symmetries of bodies in R3 41 d. Isometries of the plane 44 Lecture 5 Friday, September 11 46 a. Even and odd isometries 46 b. Isometries are determined by the images of three points 46 c. Isometries are products of reflections 48 d. Isometries in R3 50 iii iv CONTENTS e. The group structure of Isom(R2) 51 Lecture 6 Monday, September 14 54 a. Finite symmetry groups 54 b. Discrete symmetry groups 57 Lecture 7 Wednesday, September 16 61 a. Remarks on Homework #3 61 b. Classifying isometries of R3 61 c. Isometries of the sphere 64 d. The structure of SO(3) 64 e. The structure of O(3) and odd isometries 66 Lecture 8 Friday, September 18 67 a. Odd isometries with no fixed points 67 b. Remarks from Thursday tutorial: Solution to Homework #2, Exercise 11(1) 67 c. Finite subgroups of Isom(R3) 68 Lecture 9 Monday, September 21 74 a. Completion of the proof of Theorem 8.2 74 b. Regular polyhedra 76 c. Completion of classification of isometries of R3 79 d. From synthetic to algebraic: Scalar products 81 Lecture 10 Wednesday, September 23 83 a. Convex polytopes 83 b. Transformation groups and symmetries of polytopes 84 c. Regular polytopes 85 d. Back to scalar products 86 Chapter 3. Groups of matrices: Linear algebra and symmetry in various geometries 89 Lecture 11 Friday, September 25 89 a. Orthogonal matrices 89 b. Eigenvalues, eigenvectors, and diagonalizable matrices 91 c. Complexification, complex eigenvectors and rotations 93 d. Differing multiplicities and Jordan blocks 95 Lecture 12 Monday, September 28 97 a. Hermitian product and unitary matrices 97 b. Normal matrices 100 c. Symmetric matrices 103 d. Linear representations of isometries and other classes of gtransformations 103 Lecture 13 Wednesday, September 30 106 a. The projective line 106 b. The projective plane 109 c. The Riemann sphere 111 Lecture 14 Friday, October 2 113 a. Review of various geometries 113 CONTENTS v b. Affine geometry 114 c. Projective geometry 116 d. The Riemann sphere and conformal geometry 118 Lecture 15 Friday, October 9 120 a. Comments on Theorem 14.1 120 b. Hyperbolic geometry 121 Lecture 16 Monday, October 12 122 a. Ideal objects 122 b. Hyperbolic distance 122 c. Isometries of the hyperbolic plane 126 Lecture 17 Wednesday, October 14 128 a. Matrix groups 128 b. Algebraic structure of the unipotent group 129 c. The Heisenberg group 130 Lecture 18 Friday, October 16 132 a. Groups of unipotent matrices 132 b. Matrix exponentials and relationship between multiplication and addition 134 Lecture 19 Monday, October 19 137 a. Lie algebras 137 b. Lie groups 139 c. Examples 140 Chapter 4. Fundamental group: A different kind of group associated to geometric objects 143 Lecture 20 Wednesday, October 21 143 a. Isometries and homeomorphisms 143 b. Tori and Z2 144 c. Paths and loops 146 d. The fundamental group 148 e. Algebraic topology 151 Lecture 21 Friday, October 23 153 a. Homotopic maps, contractible spaces and homotopy equivalence 153 b. The fundamental group of the circle 156 c. Direct products, spheres, and bouquets of circles 157 Lecture 22 Monday, October 26 160 a. Fundamental groups of bouquets of circles 160 Chapter 5. From groups to geometric objects and back 167 Lecture 23 Wednesday, October 28 167 a. Cayley graphs 167 b. Homotopy types of graphs 169 c. Covering maps and spaces 172 d. Deck transformations and group actions 174 vi CONTENTS Lecture 24 Friday, October 30 176 a. Subgroups of free groups are free 176 b. Abelian fundamental groups 176 c. Finitely presented groups 177 d. Free products 180 Lecture 25 Monday, November 2 181 a. Planar models 181 b. The fundamental group of a polygonal complex 184 c. Factor spaces of the hyperbolic plane 186 Lecture 26 Wednesday, November 4 188 a. Hyperbolic translations and fundamental domains 188 b. Existence of free subgroups 191 c. Surfaces as factor spaces 193 Lecture 27 Friday, November 6 195 a. A rough sketch of geometric representations of surface groups 195 b. Fuchsian groups 196 c. More on subgroups of PSL(2, R) 198 d. The Heisenberg group and nilmanifolds 199 Chapter 6. Groups at large scale 203 Lecture 28 Monday, November 9 203 a. Commensurability 203 b. Growth in finitely generated groups 204 Lecture 29 Wednesday, November 11 207 a. Different growth rates 207 b. Groups with polynomial growth 208 c. Abelian groups 209 d. Nilpotent groups 210 Lecture 30 Friday, November 13 213 a. The automorphism group 213 b. Outer automorphisms 214 Lecture 31 Monday, November 16 216 a. The structure of SL(2, Z) 216 b. The space of lattices 218 c. The structure of SL(n, Z) 219 Lecture 32 Wednesday, November 18 222 a. Generators and generating relations for SL(n, Z) 222 b. Semi-direct products 224 c. Examples and properties of semi-direct products 226 Lecture 33 Friday, November 20 228 a. Quasi-isometries 228 b. Quasi-isometries and growth properties 229 c. Geometric properties of groups 230 CHAPTER 1 Elements of group theory Lecture 0. Wednesday, Thursday, and Friday, August 26–28 a. Binary operations. We learn very early on that numbers are more than just static symbols which allow us to quantify one thing or another. Indeed, they can be added, subtracted, multiplied, and (usually) divided, and this toolbox of arithmetic operations is largely responsible for the phe- nomenal variety of uses we are able to make of the concept of “number”. The study of arithmetic is the study of the behaviour of these tools and of the numbers with which they interact. Given the utility of these arithmetic tools, one may reasonably ask if their usefulness is inextricably bound up in their domain of origin, or if we might profitably apply them to something besides numbers (although just what else we ought to use in place of numbers is not obvious). Fortunately for our purposes (otherwise this course would not exist), the latter alternative turns out to be the correct one. As so often happens in mathematics, the arithmetic tools of addition, multiplication, etc. can be abstracted and generalised for use in a much broader setting; this generali- sation takes us from arithmetic to algebra, and leads us eventually into an incredibly rich field of mathematical treasures. We begin our story where so many mathematical stories begin, with a set X, which we then equip with some structure. In this case, the key structure we will place on X is a binary operation—that is, a function which takes two (possibly equal) elements of X as its input, and returns another (possibly equal) element as its output. Formally, a binary operation is a map from the direct product X X to the original set X. Rather than using the functional notation f : X X× X for this map, so that the output associated to the inputs a and b ×is written→ as f(a, b), it is standard to represent the binary operation by some symbol, such as ⋆, which is written between the two elements on which it acts—thus in place of f(a, b) we write a ⋆ b. This notation becomes intuitive if we consider the familiar examples from arithmetic. If X is some class of numbers (such as N, Z, R, or C), then the usual arithmetic operators are binary operations; for example, addition defines a map f+ : X X X by f+(a, b) = a + b, subtraction defines a map f (a, b)= a b,× and so→ on. − − 1 2 1.ELEMENTSOFGROUPTHEORY Exercise 0.1. In fact, subtraction and division only define binary op- erations on some of the classes of numbers listed above. On which sets do they fail to be binary operations, and why? When we need to indicate which binary operation a set is equipped with, we shall do so by writing the two as an ordered pair. Thus (N, +) denotes the natural numbers equipped with addition, while (R, ) denotes the real numbers with multiplication. · Since we tend to think of arbitrary binary operations as generalisations of either addition or multiplication, it is common to refer to a ⋆ b as either the “sum” or the “product” of a and b, even when (X, ⋆) is arbitrary. Example 0.1. Although addition and multiplication are the two pri- mary examples, we can define many other binary operations on the above sets of numbers, not all of which are particularly interesting: (1) a ⋆ b = ab + 2 (2) a ⋆ b = a + b 5 (3) a ⋆ b = a 3 − (4) a ⋆ b = 4 − Example 0.2.
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