Index 337
monad, 267
Newman’s first theorem, 291
Newman’s second theorem, 292
nilpotent Lie algebra, 251
nilprogression, 165
nilradical, 257
no small subgroups, 106
noncommutative progression, 18, 165
nonstandard finite set, 158
nonstandard function, 152
nonstandard object, 149
nonstandard space, 149
nonstandard universe, 149
normal Sanders lemma, 182
normal sub-local group, 119
normalising neighbourhood, 119
notation, xii
NSS, 106
NSS approximate group, 208
one-parameter subgroup, 41, 57
open mapping theorem, 56
orthogonal group, 9
outer regularity, 77
overspill principle, 155
periodic group, 233
Peter-Weyl theorem, 298
Plancherel identity, 303
Poincar´ e-Birkhoff-Witt theorem, 250
pointed Gromov-Hausdorff convergence,
174
pointed metric space, 174
probability space, 313
profinite group, 130, 134
projective limit, 93
pseudometric, 100
pushforward, 32
quotient space, 120
quotiented commutator map, 242
radially homogeneous
C1,1 local group,
39
radical, 255
Radon measure, 77
regular representation, 74, 88
restricted Burnside problem, 235
restriction of a local group, 27
Riesz lemma, 64
Riesz representation theorem, 78
Ruzsa covering lemma, 183
Ruzsa triangle inequality, 221
Sanders lemma, 168, 212
Schur’s lemma, 298
Schur’s theorem, 233
smooth action, 87
smooth atlas, 8
smooth manifold, 8
smooth structure, 8
soft analysis, 139
solenoid, 12
special orthogonal group, 9
special unitary group, 9
spectral theorem, 90
stabiliser subgroup, 86
standard function, 152
standard object, 148
standard part, 159
standard universe, 148
Stone-Cech compactification, 146
strictly nonstandard, 150
sub-local group, 119
subgroup trapping, 112, 122
submanifold, 54
syndetic set, 95
tangent bundle, 31
tangent space, 30
tangent vector, 31
Tonelli theorem, 319
topological group, 10
topological manifold, 8
topological vector space, 61
torsion group, 162
totally disconnected space, 128
transitive action, 86, 127
triangle removal lemma, 315, 321
ultra approximate group, 162, 212
ultrafilter, 145
ultralimit, 149
ultrapower, 149
ultraproduct, 149
uniform total boundedness, 172
uniqueness of Lie structure, 42
unitary group, 9
universal cover, 226
universal enveloping algebra, 249
van Dantzig’s theorem, 128
vector field, 32
von Neumann’s theorem, 54
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