title: Right-angled buildings and locally compact groups
Creators: name: Beßmann, Lara
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2024
Subjects: subject: 510 Mathematics

description: Universal groups are certain automorphism groups of right-angled buildings, that are defined via prescribed local actions. Equipped with the topology of pointwise convergence, these groups are second countable and totally disconnected locally compact groups. We present conditions that lead to a concrete description of the identity neighbourhoods, and topologically isomorphic universal groups that are associated to different right-angled buildings. Furthermore, we describe a finite presentation for discrete chamber-transitive universal groups. Automatic continuity asks for the continuity of abstract group homomorphisms between topological groups. We study this question for group homomorphisms from Polish or locally compact groups to universal groups and present conditions on the local actions that ensure continuity of abstract surjective group homomorphisms. Additionally, we obtain the uniqueness of the group topology in most cases.
description: Universelle Gruppen sind bestimmte Automorphismengruppen rechtwinklige Gebäude, die durch vorgegebene lokale Wirkungen definiert sind. Versehen mit der Topologie der punktweisen Konvergenz, sind sie zweitabzählbare, total unzusammenhängende lokalkompakte Gruppen. Neben Bedingungen, die uns eine konkrete Beschreibung der Einsumgebungen ermöglichen, präsentieren wir universelle Gruppen, die topologisch isomorph und assoziiert zu verschiedenen rechtwinkligen Gebäuden sind. Außerdem geben wir eine endliche Präsentierung für diskrete Kammern-transitive universelle Gruppen an. Automatische Stetigkeit fragt nach der Stetigkeit abstrakter Gruppenhomomorphismen zwischen topologischen Gruppen. Wir studieren diese Frage für Gruppenhomomorphismen von polnischen oder lokalkompakten Gruppen zu universellen Gruppen und geben Voraussetzungen an die lokalen Wirkungen an, sodass abstrakte surjektive Gruppenhomomorphismen stetig sind. Zusätzlich erhalten wir in den meisten Fällen die Eindeutigkeit der Gruppentopologie.
description: lntroduction ..... XIII
1 Preliminaries ..... 1
1.1 Topological spaces ..... 1
1.1.1 General topological properties ..... 2
1.1.2 Maps between topological spaces ..... 3
1.1.3 Polish spaces ..... 4
1.2 Group theory ..... 5
1.2.1 Permutation groups and group actions ..... 6
1.2.2 Topological groups ..... 8
1.3 Universal groups for trees ..... 13
1.4 Buildings and automorphisms ..... 16
1.4.1 Coxeter systems ..... 17
1.4.2 Buildings ..... 18
1.4.3 Automorphisms of buildings and universal groups ..... 26
2 Properties of universal groups ..... 35
2.1 Subgroups of universal groups ..... 35
2.1.1 Centraliser lattice for non-discrete universal groups ..... 40
2.1.2 Stabiliser of wings and their centralisers ..... 41
2.1.3 Identity neighbourhoods in universal groups ..... 49
2.2 Uniqueness of universal groups ..... 52
2.3 A finite presentation for discrete universal groups ..... 58
3 Automatie continuity for universal groups ..... 63
3.1 Non-discrete universal groups ..... 64
3.1.1 Universal groups over irreducible buildings ..... 64
3.1.2 Universal groups over reducible buildings ..... 67
3.1.3 Simple universal groups ..... 68
3.2 Products of universal groups and Polish groups ..... 70
3.3 Discrete universal groups ..... 71
Open questions and further problems ..... 73
Bibliography ..... 75
Index ..... 79