Dokument: On some algebraic properties of p-adic analytic pro-p groups

Titel:	On some algebraic properties of p-adic analytic pro-p groups
URL für Lesezeichen:	https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=65729
URN (NBN):	urn:nbn:de:hbz:061-20240510-105407-0
Kollektion:	Dissertationen
Sprache:	Englisch
Dokumententyp:	Wissenschaftliche Abschlussarbeiten » Dissertation
Medientyp:	Text
Autor:	Conte, Martina [Autor]
Dateien:	[Dateien anzeigen] Adobe PDF [Details] 1,20 MB in einer Datei [ZIP-Datei erzeugen] Dateien vom 04.05.2024 / geändert 04.05.2024
Beitragende:	Priv.-Doz. Dr. Klopsch, Benjamin [Betreuer/Doktorvater] Prof. Dr. Macpherson, H. Dugald [Gutachter]
Dewey Dezimal-Klassifikation:	500 Naturwissenschaften und Mathematik » 510 Mathematik
Beschreibung:	This thesis deals with algebraic properties of profinite groups. It is divided into two parts, corresponding to two distinct topics. The first part is devoted to proving the finite axiomatizability of the rank and the dimension of pro-π groups, while the second part is about the unique product property for pro-p groups. Recently, Nies, Segal and Tent investigated finite axiomatizability in the realm of profinite groups. They prove that the rank of a p-adic analytic pro-p group is finitely axiomatizable up to an error term. It is therefore natural to ask whether the rank of a pro-p group can be completely determined by a single first-order sentence. Here we give a positive answer to this question. More generally, given a finite set of primes π, we consider the class of pro-π groups and we prove that the rank of a pro-π group, as well as the ranks and dimensions of its Sylow pro-p subgroups, are finitely axiomatizable in the first-order language of groups. Moreover, we show that this result is optimal in the class of profinite groups. The result is first proved for the profinite groups in the class C_π of pronilpotent groups whose order is divisible only by primes in π and it is subsequently extended to pro-π groups. Its proof is based on group-theoretic results that are of independent interest. The second part of the thesis concerns the unique product property for pro-p groups. A group G has the unique product property, or equivalently G is a unique product group, if, given two non-empty, finite subsets A and B of G, there always exists at least one element g of G that can be written in a unique way as a product g = ab with a ∈ A and b ∈ B. The unique product property was introduced in 1964 by Rudin and Schneider in connection with Kaplansky’s conjecture on zero divisors in group rings. It is indeed not difficult to show that a unique product group satisfies this conjecture. Recently, Craig and Linnell conjectured that uniform pro-p groups possess the unique product property. By extending one of their results we prove that the conjecture holds true for virtually soluble saturable pro-p groups. A well-known property that is stronger than the unique product property is local indicability. A group is locally indicable if each of its non-trivial finitely generated subgroups has infinite abelianisation. We start to study local indicability for soluble profinite groups, producing some results that relate being locally indicable to a topological version of local indicability. Another property related to local indicability and the unique product property is orderability. Indeed, one can show that a bi-orderable group is locally indicable. We give an elementary proof of the fact that insoluble pro-p groups of finite rank are not bi-orderable and we adapt one of the proofs that RAAGs are bi-orderable to show that also pro-p completions of RAAGs are bi-orderable, hence locally indicable.
Lizenz:	Dieses Werk ist lizenziert unter einer Creative Commons Namensnennung 4.0 International Lizenz
Fachbereich / Einrichtung:	Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Algebra und Zahlentheorie
Dokument erstellt am:	10.05.2024
Dateien geändert am:	10.05.2024
Promotionsantrag am:	26.10.2023
Datum der Promotion:	24.01.2024