This second volume in a two-volume set provides a complete self-contained proof of the classification of geometries associated with sporadic simple groups: Petersen and tilde geometries. It contains a study of the representations of the geometries under consideration in GF(2)-vector spaces as well as in some non-Abelian groups. The central part is the classification of the amalgam of maximal parabolics, associated with a flag transitive action on a Petersen or tilde geometry. By way of their systematic treatment of group amalgams, the authors establish a deep and important mathematical result.
Author(s): A. A. Ivanov, S. V. Shpectorov
Edition: 1
Year: 2002
Language: English
Pages: 304
Cover......Page 1
Encyclopedia of Mathematics and its Applications 91......Page 2
Geometry of Sporadic Groups II: Representations and Amalgams......Page 4
9780521623490......Page 5
Contents......Page 6
Preface......Page 10
1.1 Geometries and diagrams ......Page 20
1.2 Coverings of geometries ......Page 22
1.3 Amalgams of groups ......Page 24
1.4 Simple connectedness via universal completion ......Page 26
1.5 Representations of geometries ......Page 30
Part I. Representations ......Page 36
2.1 Terminology and notation ......Page 38
2.2 Collinearity graph ......Page 41
2.3 Geometric hyperplanes ......Page 43
2.4 Odd order subgroups ......Page 46
2.5 Cayley graphs ......Page 51
2.6 Higher ranks ......Page 53
2.7 c-extensions ......Page 54
2.8 Non-split extensions ......Page 59
3.1 Linear groups ......Page 62
3.2 The Grassmanian ......Page 64
3.3 \mathscr{P}^1_e is uniserial ......Page 66
3.4 \mathscr{G}(S_4(2)) ......Page 69
3.5 Symplectic groups ......Page 70
3.6 Orthogonal groups ......Page 74
3.7 Brouwer's conjecture ......Page 75
3.8 \mathscr{G}(3 \cdot S_4(2)) ......Page 83
3.9 \mathscr{G}(Alt_5) ......Page 85
3.10 \mathscr{G}(3^{\binom{n}{2}_2} \cdot S_{2n}(2)) ......Page 87
4.1 \mathscr{G}(M_{23}) ......Page 95
4.2 \mathscr{G}(M_{22}) ......Page 96
4.3 \mathscr{G}(M_{24}) ......Page 100
4.4 \mathscr{G}(3 \cdot M_{22}) ......Page 101
4.5 \mathscr{D}(M_{22}) ......Page 107
4.6 \mathscr{G}(He) ......Page 111
5.1 Leech lattice ......Page 112
5.2 \mathscr{G}(Co_2) ......Page 116
5.3 \mathscr{G}(Co_1) ......Page 118
5.4 Abelianization ......Page 120
5.5 \mathscr{G}(3^{23} \cdot Co_2) ......Page 122
5.6 \mathscr{G}(3 \cdot U_4(3)) ......Page 127
6.1 General methods ......Page 130
6.2 \mathscr{I}(Alt_7) ......Page 134
6.3 \mathscr{I}(M_{22}) ......Page 136
6.4 \mathscr{I}(U_4(3)) ......Page 139
6.5 \mathscr{I}(Co_2, 2B) ......Page 141
6.6 \mathscr{I}(Co_1, 2A) ......Page 144
7.1 Existence of the representations ......Page 147
7.2 A reduction via simple connectedness ......Page 150
7.3 The structure of N(p) ......Page 153
7.4 Identifying R_1(p)......Page 160
7.5 R_1(p) is normal in R \lfloor p \rfloor......Page 165
7.6 R \lfloor p \rfloor is isomorphic to \tilde{G}(p)......Page 170
7.7 Generation of \tilde{G}(p) \cap \tilde{G}(q)......Page 172
7.8 Reconstructing the rank 3 amalgam ......Page 174
7.9 \mathscr{G}(3^{4371} \cdot BM) ......Page 178
Part II. Amalgams ......Page 180
8.1 General strategy ......Page 182
8.2 Some cohomologies ......Page 184
8.3 Goldschmidt's theorem ......Page 189
8.4 Factor amalgams ......Page 192
8.5 L_3(2)-lemma......Page 194
8.6 Two parabolics are sufficient ......Page 197
9.1 A graph theoretical setup ......Page 199
9.2 Normal series of the vertex stabilizer ......Page 202
9.3 Condition (*_i)......Page 206
9.4 Normal series of the point stabilizer ......Page 210
9.5 Pushing up ......Page 215
10.1 The setting ......Page 217
10.2 Rank three case ......Page 219
10.3 Rank four case ......Page 223
10.4 Rank five case ......Page 228
10.6 The symplectic shape ......Page 230
10.7 Summary ......Page 232
11.1 M_{22}-shape......Page 234
11.2 Aut M_{22}-shape......Page 236
11.3 M_{23}-shape......Page 238
11.4 Co_2-shape......Page 240
11.5 J_4-shape......Page 246
11.6 Truncated J_4-shape......Page 252
11.7 BM-shape ......Page 253
12.1 Alt_7-shape......Page 261
12.2 S_6(2)-shape......Page 262
12.3 M_{24}-shape......Page 265
12.4 Truncated M_{24}-shape......Page 267
12.5 The completion of \mathscr{A}_f ......Page 272
12.6 Co_1-shape......Page 275
12.8 S_{2n}(2)-shape, n >= 4......Page 280
Concluding Remarks ......Page 288
13.1 Group-free characterizations ......Page 290
13.2 Locally projective graphs ......Page 294
References......Page 297
Index......Page 304
