Downward permanence of the strong topological Rokhlin property

Determine whether the strong topological Rokhlin property passes from a finitely generated finite-index overgroup to its finite-index subgroup, thereby establishing invariance of the strong topological Rokhlin property under commensurability.

Background

The paper proves that the strong topological Rokhlin property passes from a finite-index subgroup to a finitely generated overgroup. This establishes one direction of the permanence problem and implies that every finitely generated virtually free group, including every virtually cyclic group, has the strong topological Rokhlin property.

The converse finite-index direction—whether the property passes from a finitely generated group to a finite-index subgroup—would yield full invariance under commensurability. The paper explicitly states that this downward direction is unresolved.

References

The downward direction needed for full commensurability invariance remains open.

— Cayley-tree pseudo-orbit tracing: period subgroups and relative geometry  (2608.16483 - Xu, 17 Aug 2026) in Section 1, subsection “Cohomological dimension and generic Cantor actions”