312 Index

deck group, 66, 71

deck transformation, 66

Dehn twist, 235

Dehn’s Dissection Theorem, 17, 98,

287

determinant, 30

diffeomorphism, 135

differential representation, 223

disk model, 125

disk rigidity, 177

disk-like sets, 143, 144

dodecahedron, 63

dot product, 22, 87

double octagon, 232

edges, 5

elliptic isometry, 130

equivalence class, 32

equivalence relation, 32

essential curves, 5

Euclidean cone, 207

Euclidean cone surface, 208

Euclidean plane, 1

Euclidean space, 1, 87

Euler characteristic, 37

exponential function, 175

faces, 5

Farey graph, 242, 245

Farey related, 242

figure 8, 69

flat cone surface, 207

flat torus, 16, 67, 195, 252

functoriality, 49

fundamental group, 43, 46, 71

Fundamental Theorem of Algebra,

57, 180

Gauss map, 239

Gauss–Bonnet Theorem, 107, 127

Gauss-Bonnet Theorem, 151

generalized circle, 116

geodesic, 145

geodesic half-plane, 150

geodesic hyperbolic triangle, 228

geodesic polygon, 127, 145

geodesic triangle, 107

geodesic triangulation, 150

geodesics, 9, 105, 120, 123

Giraud’s Theorem, 107

gluing, 2, 3, 31, 145

gluing recipe, 147

good metric space, 80

Green’s Theorem, 100, 166

group, 43

Hadamard’s Theorem, 154

harmonic function, 179

holomorphic function, 163

homeomorphism, 25

homomorphism, 44

homotopy, 45

homotopy equivalence, 45

Hopf–Rinow Theorem, 154

hyperbolic geometry, 9

hyperbolic group actions, 224

hyperbolic isometry, 121, 130, 143

hyperbolic plane, 9, 224

hyperbolic reflections, 229

hyperbolic surface, 143, 147, 195

ideal triangle, 127, 242

ideal vertices, 127

Implicit Function Theorem, 28

infimum, 31

infinite differentiability, 169

inner product, 121, 134

interior, 23

Invariance of Domain, 189

Inverse Function Theorem, 135, 185

irrational numbers, 247

isometry, 135

isomorphism, 44

isomorphism theorem, 71, 74, 79

Jordan curve, 199

Jordan domain, 199

Klein bottle, 35

latitude, 105

lens space, 59

Lie group, 30

lifting property, 73

line integrals, 98

linear fractional transformation, 11,

115, 130

linear functionals, 98