On contraction groups of automorphisms of totally disconnected locally compact groups
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.
scientific article; zbMATH DE number 5610593
scientific article