viii Contents §4.1. Haar measure 76 §4.2. The Peter-Weyl theorem 87 §4.3. The structure of locally compact abelian groups 94 Chapter 5. Building metrics on groups, and the Gleason-Yamabe theorem 99 §5.1. Warmup: the Birkhoff-Kakutani theorem 101 §5.2. Obtaining the commutator estimate via convolution 103 §5.3. Building metrics on NSS groups 106 §5.4. NSS from subgroup trapping 112 §5.5. The subgroup trapping property 114 §5.6. The local group case 117 Chapter 6. The structure of locally compact groups 125 §6.1. Van Dantzig’s theorem 127 §6.2. The invariance of domain theorem 131 §6.3. Hilbert’s fifth problem 133 §6.4. Transitive actions 136 Chapter 7. Ultraproducts as a bridge between hard analysis and soft analysis 139 §7.1. Ultrafilters 144 §7.2. Ultrapowers and ultralimits 148 §7.3. Nonstandard finite sets and nonstandard finite sums 157 §7.4. Asymptotic notation 158 §7.5. Ultra approximate groups 161 Chapter 8. Models of ultra approximate groups 167 §8.1. Ultralimits of metric spaces (Optional) 169 §8.2. Sanders-Croot-Sisask theory 176 §8.3. Locally compact models of ultra approximate groups 184 §8.4. Lie models of ultra approximate groups 190 Chapter 9. The microscopic structure of approximate groups 197 §9.1. Gleason’s lemma 204 §9.2. A cheap version of the structure theorem 208 §9.3. Local groups 212 Chapter 10. Applications of the structural theory of approximate groups 219
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