A note concerning a tidying procedure and contraction groups in non-metrizable, totally disconnected groups

TL;DR

This paper extends the understanding of contraction groups in totally disconnected, locally compact groups to the non-metrizable case, addressing a gap in previous work and confirming the validity of earlier results.

Contribution

It resolves a key technical difficulty in applying contraction group theory to non-metrizable groups, affirming the generality of prior results.

Findings

Confirmed the validity of contraction group results in non-metrizable groups

Resolved a technical gap in the theory of totally disconnected groups

Extended the applicability of structure theory to broader classes of groups

Abstract

In a 2004 article, Udo Baumgartner and George Willis used ideas from the structure theory of totally disconnected, locally compact groups to achieve a better understanding of the contraction group U_f associated with an automorphism f of such a group G, assuming that G is metrizable. (Recall that U_f consists of all group elements x such that f^n(x) tends to the identity element as n tends to infinity). Recently, Wojciech Jaworski showed that the main technical tool of the latter article remains valid in the non-metrizable case. He asserted without proof that, therefore, all results from that article remain valid. However, metrizability enters the arguments at a second point. In this note, we resolve this difficulty, by providing an affirmative answer to a question posed by Willis in 2004.