Zimmer Rigidity for Homeomorphism Actions

Definitions and mathematical statement

An action of a group $\Gamma$ by homeomorphisms on a topological manifold $M$ is a homomorphism $\rho\colon\Gamma\to\operatorname{Homeo}(M)$, where $\operatorname{Homeo}(M)$ is the group of self-homeomorphisms of $M$. A closed manifold is compact and has no boundary. Call the action virtually trivial if there is a subgroup $\Gamma_0\leq\Gamma$ such that \[ [\Gamma:\Gamma_0]<\infty \quad\text{and}\quad \rho(\gamma)=\operatorname{id}_M\quad\text{for every }\gamma\in\Gamma_0. \] Equivalently, $\rho(\Gamma)$ is finite.

The supplied problem asks whether the dimension-dependent rigidity assertion in Zimmer's conjecture continues to hold for actions by homeomorphisms. Its intended higher-rank lattice class and dimension bound are not specified. To exhibit the quantifiers without supplying missing hypotheses, let $\mathcal H$ denote the unspecified intended class of pairs $(G,\Gamma)$, where $\Gamma$ is a higher-rank lattice in an ambient group $G$, and let $b(G,\Gamma)$ denote the unspecified prescribed dimension threshold. These symbols are placeholders, not additional mathematical data supplied by the source. The question has the schematic form: \[ \begin{gathered} \text{For every }(G,\Gamma)\in\mathcal H,\text{ every closed topological manifold }M,\\ \text{and every homomorphism }\rho\colon\Gamma\to\operatorname{Homeo}(M),\\ \dim M<b(G,\Gamma) \quad\Longrightarrow\quad \lvert\rho(\Gamma)\rvert<\infty\;? \end{gathered} \] Here the strict inequality records the source's phrase "below the prescribed dimension bound"; no numerical threshold or endpoint case is supplied. The ambient groups, lattice hypotheses, meaning of higher rank in the intended formulation, and any additional manifold or action hypotheses remain unresolved. Consequently, this is a quantified template for the supplied question, not a fully specified conjecture.

Short English statement

Must every homeomorphism action of a higher-rank lattice on a closed manifold below the intended Zimmer dimension threshold become trivial on a subgroup of finite index? The supplied record leaves the lattice class and threshold unspecified, so a precise conjecture requires further source clarification.

Sources

• Local UnsolvedMath registry: permanent ID $30003837$, title Zimmer Rigidity for Homeomorphism Actions, exact source problem code catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2018, and names no proposer.
• Supplied source report: Cohomological and Metric Properties of Groups of Homeomorphisms of R (2019), Oberwolfach Reports, pp. 1579--1633, DOI 10.4171/owr/2018/26. https://doi.org/10.4171/owr/2018/26. The background attributes the collection to workshop reports and their open-problem discussions.
• Supplied literature reference: Zimmer program survey. https://doi.org/10.1090/noti2252.
• Supplied literature reference: Lodha--Matte Bon--Triestino. https://doi.org/10.1112/topo.12151.
• Supplied literature reference: Oberwolfach Report 2019. https://ems.press/content/serial-article-files/46748?nt=1.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that Zimmer-type rigidity remains open for homeomorphism actions in the $C^0$ setting despite substantial smooth and singular-diffeomorphism results. This is supplied dated curation derived from the OWR material, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry supplies statement status corrected_verified and reports manual semantic, typographic, and LaTeX review asserting that the clean statement is self-contained and preserves the mathematical target. That supplied review label does not resolve the missing hypotheses or threshold identified here.