title: Locally compact groups, automatic continuity and geometric group theory
Creators: name: Möller, Philip
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2024
Subjects: subject: 510 Mathematics

description: Wir zeigen, dass eine gutartige Wirkung einer lokalkompakten Gruppe auf einem CAT(0) Raum stetig ist, oder einen invarianten Unterraum hat. Außerdem zeigen wir, dass Homomorphismen von lokalkompakten Gruppen in „geometrische“ Gruppen entweder stetig sind oder „kleines“ Bild haben.
Für pro-Liegruppen zeigen wir, dass gutartige Einbettungen „fast“ eine Zerlegung in ein direktes Produkt induzieren.
Wir charakterisieren (Semi-)Vollständigkeit eines Graphprodukts anhand des Graphen und untersuchen normale Untergruppen.
Wir zeigen, dass Moussong's Charakterisierung von Gromov-hyperbolischen Coxetergruppen eine Lücke hat und schließen diese. 
Danach zeigen wir, dass eine Coxetergruppe genau dann „just-infinite“ ist, wenn sie irreduzibel und affin ist. Wir zeigen diverse Resultate über proendliche Starrheit von Graphprodukten und Coxetergruppen. 
In Artingruppen zeigen wir, dass der Schnitt einer vollständigen parabolischen Untergruppe mit einer parabolischen Untergruppe parabolisch ist.
description: 1 Introduction ..... 1
1.1 Automatic continuity ..... 1
1.2 Locally compact groups ..... 5
1.3 Automorphisms of graph products ..... 7
1.3.A Semicompleteness and completeness ..... 7
1.3.B Normal subgroups ..... 9
1.3.C The center of the automorphism group of a graph product ..... 10
1.4 Coxeter groups and Artin groups ..... 11
1.4.A Coxeter groups and hyperbolicity ..... 11
1.4.B Coxeter groups, profinite rigidity and just infinity ..... 12
1.4.C Artin groups ..... 16
2 Preliminaries ..... 19
2.1 Locally compact groups ..... 20
2.1.A General theory ..... 20
2.1.B Torsion and divisible groups ..... 25
2.1.C Abelian divisible groups ..... 26
2.1.D Pro-Lie groups ..... 27
2.2 Graph products and Coxeter/Artin groups ..... 30
2.2.A Simplicial graphs ..... 30
2.2.B Basics on the groups ..... 33
2.2.C Retractions and amalgamation ..... 35
2.2.D Irreducibility and affine Coxeter groups ..... 38
2.2.E Separability ..... 43
2.2.F The poset of conjugacy classes of finite subgroups ..... 44
2.3 The automorphism group of a graph product ..... 45
2.3.A Generators of the automorphism group ..... 46
2.3.B Subgroups of graph products ..... 47
2.4 Profinite rigidity ..... 49
2.4.A Basics ..... 49
2.4.B Profinite completions and semidirect products ..... 51
2.4.C Profinite rigidity and graph products ..... 52
2.5 Metric spaces and geometry ..... 53
2.5.A CAT(0) geometry ..... 54
2.5.B Metrically injective spaces ..... 58
3 Automatic Continuity ..... 61
3.1 The geometric perspective ..... 61
3.1.A Compact and divisible groups ..... 61
3.1.B The proof of Theorem 3.1.6 ..... 64
3.1.C Application ..... 69
3.2 The algebraic perspective ..... 74
3.2.A Virtually lcH -slender groups ..... 74
3.2.B Proof of Theorem 3.2.5 ..... 75
3.2.C On discrete groups with properties (i)-(iii) of Theorem 3.2.5 ..... 79
4 Locally compact groups and reductivity ..... 87
4.1 Compact connected groups ..... 87
4.2 Connected locally compact groups ..... 94
4.2.A The abelian case ..... 94
4.2.B The general case and different versions of reductivity ..... 95
5 Automorphisms of graph products ..... 101
5.1 Semicomplete and complete graph products of groups ..... 101
5.1.A Semicompleteness ..... 101
5.1.B Completeness ..... 106
5.2 Normal subgroups of Aut(GΓ) ..... 109
5.2.A Full-sized normal subgroups ..... 109
5.2.B Normal subgroups in automorphism groups ..... 112
5.2.C Characteristic subgroups ..... 113
5.2.D The automorphism group of a graph product of cyclic groups ..... 118
5.2.E Proof of Theorem 5.2.23 ..... 120
5.2.F Proof of Theorem 5.2.25 ..... 123
5.3 The center of the automorphism group of a graph product ..... 125
5.3.A The center of the automorphism group of an infinite finitely generated abelian group with torsion ..... 128
5.3.B The center of the automorphism group of a non-abelian graph product of cyclic groups ..... 132
5.4 Applications ..... 137
6 Coxeter groups and Artin groups ..... 141
6.1 Coxeter groups and Gromov-Hyperbolicity ..... 141
6.1.A Almost negative matrices, nerve-complexes and the Davis complex ..... 141
6.1.B The gap in the original argument ..... 144
6.1.C The characterization of hyperbolic Coxeter groups ..... 152
6.2 Irreducible affine Coxeter groups ..... 160
6.2.A Just infinity ..... 160
6.2.B Proof of Theorem 6.2.11 ..... 163
6.2.C Profinite rigidity of the infinite Dihedral group ..... 166
6.3 Profinite rigidity and Higman-Thompson groups ..... 167
6.3.A Profinite rigidity of graph products and RACGs ..... 167
6.3.B A diversion: Higman-Thompson groups Vn ..... 170
6.3.C Grothendieck pairs and flexibility ..... 176
6.4 Parabolic subgroups of Artin groups ..... 182
6.4.A Parabolic subgroups of Coxeter and Artin groups ..... 182
6.4.B Bass-Serre theory meets Artin groups ..... 186
6.4.C The clique-cube complex ..... 193
7 Open questions and future research ..... 199
7.1 Automatic continuity ..... 199
7.2 Locally compact groups and reductivity ..... 201
7.3 Automorphism groups of graph products ..... 201
7.4 Coxeter groups and Artin groups ..... 202
7.4.A Profinite rigidity ..... 202
7.4.B Artin groups ..... 204