Finite PL/projective actions for G_t of types F and T

Determine whether the forest‑skein groups G_t of type F and type T (constructed from the skein presentation (G)) admit any non‑trivial finite piecewise linear or finite piecewise projective actions on a 1‑manifold (e.g., on R or on the circle R/Z).

Background

The authors prove that the (J)‑type groups have no non‑trivial finite PL or finite piecewise projective actions on the circle. For the (G) family, they show G_t does not contain non‑abelian free subgroups and raise the question whether the F‑type or T‑type versions of G_t admit any finite PL or projective actions.

Resolving this would clarify whether the lack of finite PL/projective actions is specific to the (J) construction or extends more broadly to other forest‑skein families, and it would refine the understanding of how the algebraic structure of G_t constrains its possible smooth or piecewise dynamics.