title: Automatic continuity in the Čech complete setting
Creators: name: Keppeler, Daniel
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2024
Subjects: subject: 510 Mathematics

description: Gegeben seien zwei topologische Gruppen G und H und ein Homomorphismus von G nach H. Eine natürliche Frage ist, ob der Homomorphismus nicht nur die algebraische Struktur, sondern auch die topologische Struktur der Gruppen erhält, also stetig ist. In dieser Arbeit betrachten wir den Fall, wo G Cech vollständig und H eine diskrete und geometrisch interessante Gruppe ist. Wir definieren Cech vollständig slender Gruppen und zeigen, dass viele Gruppen, z.B. frei (abelsche) Gruppen, diese Eigenschaft besitzen. Dies nutzen wir und zeigen, dass (äußere) Automorphismengruppen von rechtwinkligen Artingruppen virtuell Cech vollständig slender sind. Weiter zeigen wir, dass jeder Homomorphismus von einer Cech vollständigen Gruppe zu einem Graphprodukt stetig ist oder ein kleines Bild besitzt. Ähnliches erhalten wir für Homomorphismen von lokalkompakten Gruppen zu Gruppen, welche weder die rationalen Zahlen, noch die p-adischen Zahlen, noch die Prüfer-p-Gruppen enthalten und deren torsions Untergruppen klein sind.
description: Given two topological groups G and H and an abstract homomorphism from G to H a natural question is if the homomorphism not only preserves the algebraic group structure, but also the topological one, i.e. if it is continuous. In this thesis, we consider the case where G is a Cech complete group and H is a discrete geometrically interesting group. We introduce the notion of Cech complete slender groups and show that many groups, e.g. free (abelian) or finitely generated torsion-free nilpotent groups, satisfy this notion. Concluding from this, we show that (outer) automorphism groups of right-angled Artin groups are virtually Cech complete slender. Further, we show that homomorphisms from Cech complete groups to graph products are automatically continuous or their image is small, i.e. lies in a complete parabolic subgroup of the graph product. Replacing the group G by some locally compact group, we conclude similar automatic continuity results to general geometric groups whose torsion subgroups are small.
description: 1 Introduction ..... 1
2 Preliminaries ..... 9
2.1 Topological spaces ..... 9
2.2 Group theory ..... 13
2.3 Topological groups ..... 23
2.4 Graph products ..... 30
3 ˇCech complete slender groups ..... 37
3.1 Automatic continuity and slenderness in various settings ..... 37
3.2 ˇCech complete slenderness ..... 40
3.2.1 Principal congruence subgroups have finite roots ..... 44
3.2.2 ˇCech complete slender abelian groups ..... 45
4 Automatic continuity for groups with torsion ..... 49
4.1 Virtually slender groups ..... 49
4.2 Automatic continuity for groups whose torsion subgroups are small ..... 51
4.2.1 On discrete groups with properties (i)-(iii) of Theorem 4.2.2 ..... 54
4.2.2 Geometric groups ..... 56
4.2.3 The class G and graph products ..... 60
4.3 Automatic continuity for acylindrically hyperbolic groups ..... 62
5 Automatic continuity for graph products ..... 69
5.1 Automatic continuity for the infinite dihedral group ..... 69
5.2 Automatic continuity for graph products ..... 75
5.3 Free products and other applications ..... 83
6 Final remarks and related questions ..... 89
Bibliography ..... 93