Closed Contraction Groups of Totally Disconnected Groups

Definitions and mathematical statement

Let $G$ be a totally disconnected locally compact Hausdorff topological group, with identity $1$, and let $\alpha:G\to G$ be a topological automorphism. Its contraction group is \[ U_\alpha=\{x\in G:\alpha^n(x)\longrightarrow1\text{ as }n\to\infty\}. \] Explicitly, $x\in U_\alpha$ means that for every neighborhood $V$ of $1$, there exists $N\geq0$ such that $\alpha^n(x)\in V$ for every integer $n\geq N$.

The question is to identify conditions on $(G,\alpha)$ under which $U_\alpha$ is closed in the given topology of $G$, that is, \[ \overline{U_\alpha}^{\,G}=U_\alpha. \] It is a request for criteria or a characterization, not an assertion that every contraction group is closed. No second-countability, metrizability, linearity, compact-generation, or building-action hypothesis is stated. Accordingly closedness is understood topologically, rather than being reduced without justification to a criterion using only sequences. The extraction gives no particular proposed necessary-and-sufficient condition or distinguished subclass whose case is the intended remaining problem.

Short English statement

Which totally disconnected locally compact groups and automorphisms have a contraction subgroup that is closed in the original group topology?

Sources

• Local UnsolvedMath registry, permanent ID $30000914$, source code catalogue entry, "Closed Contraction Groups of Totally Disconnected Groups." No individual proposer is supplied.
• Buildings: Interactions with Algebra and Geometry (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/03, pp. 119--172: https://doi.org/10.4171/owr/2008/03. This is the sole supplied external reference.
• Supplied literature triage, checked 2026-08-21: status "open." It describes the extraction as a fragment referring to an unstated result and closure question, and says no resolutive theorem can be verified from the record. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.