Index 1-cocycle, 105 2-coboundary, 101 2-cocycle, 100 2q-conditions, 265 A-acyclic, 48 CG(K), 75 E-orbits, 299 E(3), 86 F -orbits, 299 G-equivalence, 2 G-invariant tubular neighborhood of Gx, 12 G-isomorphism, 2 G-map, 1 G-space, 1 G × (f, e) Q, 96 H-kernel, 11 H-slice, 11 H1(Q C), 105 H2(Q A), 100 H2(Q Z(G)), 104 K-manifold, 56 aff(G), 231 Aff(M), 249 Aut(Π, Γ), 221 Aut0(G, K), 160 Q-acyclic, 48 E(M), 257 Z(G), 75 ℓ(G), 75 Endo(G), 231 L-group, 209 Zp-acyclic, 48 A-admissible, 56 MG(P,G), 74 μ, 75 μ(a), 75 Fitt(Γ), 153 Opext(Q,G, e), 97 Out(G), 75 Out(Π, Γ), 221 TOPG(P ), 73 p2-conditions, 265 pq-condition, 265 r(a), 76 L+-group, 209 TOP(X), 2 (S1), 122 (S2), 122 (S3), 122 (S4), 122 KAN decomposition, 117 Sol-geometry, 147 covering space construction, 23 abstract kernel, 95, 217 ACG, 227 action effective, 3 free, 4 ineffective, 3 ineffective part, 3 injective, 50 injective torus, 66 lifting exact sequence, 27 locally proper, 4 proper, 5 simply transitive, 3 smooth, 2 transitive, 3 admissible group, 217 admissible space, 54, 57 affine crystallographic group, 227 affine diffeomorphism, 82 affine endomorphisms, 231 affine structure, 205 affine structure on Γ, 227 algebraic hull, 230 almost Bieberbach group, 88 almost crystallographic group, 88 almost flat, 88 aspherical, 47, 56 base space, 6 Bieberbach group, 86, 139 almost, 88 Bieberbach Property, 115 Bieberbach Theorem 393

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