Metrizable TAP, HTAP and STAP groups
type: Item
instance of: scholarly article
title: Metrizable TAP, HTAP and STAP groups
zbMATH Open document ID: 1258.22001
published in: Topology and its Applications
publication date: 2012-06-19T00:00:00Z
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 43A55
Mathematics Subject Classification ID: 22A10
Mathematics Subject Classification ID: 46A19
zbMATH DE Number: 6047960
zbMATH Keywords: Topological group
zbMATH Keywords: NSS group
zbMATH Keywords: Multipliable sequence
zbMATH Keywords: Summable sequence
zbMATH Keywords: TAP
zbMATH Keywords: STAP
zbMATH Keywords: HTAP
zbMATH Keywords: Complete
zbMATH Keywords: locally root invariant
author: X. Domínguez
author: V. Tarieladze
reviewed by: Jorge Galindo
MaRDI profile type: Publication
full work available at URL: j.topol.2011.09.046
OpenAlex ID: W2164757228
cites work: Q3265048
cites work: Q4104968
cites work: Productivity of sequences with respect to a given weight function
cites work: Group topologies on vector spaces and character lifting properties
cites work: Remarks on independent sequences and dimension in topological linear spaces
cites work: Group-valued continuous functions with the topology of pointwise convergence
DOI: 10.1016/J.TOPOL.2011.09.046
Recommended article: Metrizability of topological groups
Recommended article: Transmetabelian groups
Recommended article: Metrical universality for groups
Recommended article: Metacyclic groups
Recommended article: Metrizability of CHART groups
Recommended article: Q4212257
Recommended article: Metanilpotent varieties of groups
Recommended article: Groups, measures, and the NIP
Recommended article: Character Tables and Metabelian Groups
review text: The authors analyze three classes of topological groups (TAP, HTAP and STAP) defined through the nonexistence of sequences with strong convergence properties that are related to the class NSS of groups with no small subgroups. The classes HTAP and STAP are introduced in the present paper as natural strengthenings of the TAP (after \textit{trivially absolutely productive}) property introduced by \textit{D. Shakhmatov} and \textit{J. Spěvák} [Topology Appl. 157, No. 8, 1518--1540 (2010; Zbl 1195.54040)].NEWLINENEWLINEA topological group is said to be a TAP group if for every faithfully indexed sequence \((g_n)_{n\in \mathbb{N}}\) of elements of \(G\) there is a sequence \((m_k)_{n\in \mathbb{N}}\) of integers and a bijective map \(\sigma: \mathbb N\to\mathbb N\) such that the sequence \(\left(\prod_{k=1}^ng_{\sigma(k)}^{m_k}\right)_{n\in \mathbb{N}}\) is not convergent, i.e., in the terminology of the present paper, if no faithfully indexed sequence is unconditionally hyper-multipliable. For a topological group to be in HTAP it is necessary that for every faithfully indexed sequence \((g_n)_{n\in \mathbb{N}}\) of elements of \(G\) there is a sequence \((m_k)_{n\in \mathbb{N}}\) of integers such that the sequence \(\left(\prod_{k=1}^n g_{k}^{m_k}\right)_{n\in \mathbb{N}}\) is not convergent. For a topological group to be in STAP it is required that for every faithfully indexed sequence \((g_n)_{n\in \mathbb{N}}\) of elements of \(G\) there is a sequence \((m_k)_{n\in \mathbb{N}}\) of integers such that the sequence \(\left(g_n^{m_n}\right)_{n\in \mathbb{N}}\) is not convergent. It is clear from the definitions that \(\mathrm{STAP}\subset \mathrm{HTAP}\subset \mathrm{TAP}\).NEWLINENEWLINEShakhmatov and Spěvák [loc. cit.] proved that \(\mathrm{NSS}\subset \mathrm{TAP}\). In the present paper it is proved that NSS groups even have the stronger STAP property (hence \(\mathrm{NSS}\subset \mathrm{STAP}\subset \mathrm{HTAP}\subset \mathrm{TAP}\)) and that for metrizable group…
description: scientific article; zbMATH DE number 6047960