On contraction groups of automorphisms of totally disconnected locally compact groups
type: Item
instance of: scholarly article
title: On contraction groups of automorphisms of totally disconnected locally compact groups
zbMATH Open document ID: 1179.22004
author: Wojciech Jaworski
published in: Israel Journal of Mathematics
publication date: 2009-10-02T00:00:00Z
review text: Let \(G\) be a topological group, with neutral element \(1\). To each automorphism \(\tau: G\to G\), a subgroup \(C(\tau)\) (the `contraction group') can be associated, which consists of all group elements \(x\in G\) such that \(\tau^n(x)\to 1\) as \(n\to\infty\). More generally, given a closed subgroup \(H\subseteq G\) such that \(\tau(H)=H\), one can consider the set \(C(\tau,H)\) of all \(x\in G\) such that \(\tau^n(x)H\to H\) in \(G/H\). The author shows that \(C(\tau,H)=C(\tau)H\) for each automorphism \(\tau\) of a totally disconnected, locally compact group \(G\) and a \(\tau\)-stable closed subgroup \(H\) of \(G\) (Theorem 1). Previously, this was only known under the additional hypothesis that \(G\) is metrizable [see \textit{U. Baumgartner} and \textit{G. A. Willis}, Israel J. Math. 142, 221--248 (2004; Zbl 1056.22001)]. In the cited paper, contraction groups were linked to the structure theory of totally disconnected groups [as initiated in \textit{G. A. Willis}, Math. Ann. 300, 341--363 (1994; Zbl 0811.22004)], in the metrizable case. For example, the closedness of \(C(\tau)\) was characterized there in many ways, and it was shown that the scale of \(\tau\) coincides with the module of \(\tau\) on the closure of the contraction group of \(\tau^{-1}\). As the author points out, all of these results remain true for not necessarily metrizable groups, as a consequence of his Theorem 1 just described.
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 22D45
zbMATH DE Number: 5610593
zbMATH Keywords: contraction group
zbMATH Keywords: automorphism
zbMATH Keywords: totally disconnected group
zbMATH Keywords: locally compact group
zbMATH Keywords: invariant subgroup
zbMATH Keywords: stable set
zbMATH Keywords: metrizability
MaRDI profile type: Publication
full work available at URL: s11856-009-0059-0
OpenAlex ID: W2009070411
cites work: Contraction groups and scales of automorphisms of totally disconnected locally compact groups
cites work: CONTRACTION GROUPS FOR TIDY AUTOMORPHISMS OF TOTALLY DISCONNECTED GROUPS
cites work: Q3205756
DOI: 10.1007/S11856-009-0059-0
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description: scientific article; zbMATH DE number 5610593