Existence of FS groups with no C1 actions and no proper CAT(0) cube complex actions

Determine whether there exist forest‑skein groups that, like Lodha’s finitely presented simple group, admit no C1 actions on any 1‑manifold and cannot act properly on a CAT(0) cube complex. Identify and construct forest‑skein groups exhibiting both properties, or show that none exist.

Background

The paper discusses Lodha’s simple group, which has no finite PL actions on the circle, does not admit C1 actions on any 1‑manifold, and (via an embedded Baumslag–Solitar subgroup) cannot act properly on a CAT(0) cube complex. The authors contrast this with their forest‑skein constructions and note a lack of examples among forest‑skein groups sharing these specific properties.

Determining whether any forest‑skein groups have both features would broaden the catalogue of dynamical and geometric behaviours within the forest‑skein framework and potentially connect it to constraints known in geometric group theory for CAT(0) cube complexes and smooth dynamics.