Obstruction to finite PL/projective actions for FS groups from (G) and (H)

Ascertain whether there exists an obstruction, analogous to the Brin–Squier theorem combined with Ghys and Fournier‑Facio–Lodha bounded cohomology arguments used for the (J) skein presentation, that forbids non‑trivial finite piecewise linear or finite piecewise projective actions on the circle for forest‑skein groups constructed from the skein presentations (G) and (H). The skein presentation (G) is FS a,b|t(a) = ρ(b), where t is a non‑trivial monochromatic tree and ρ is a right‑vine of appropriate length; the skein presentation (H) is FS i : 1 ≤ i < k | Y(i)(I ⊗ Y(j)) = Y(j)(Y(i) ⊗ I) : i < j, whose F‑type group is the k‑ary Higman–Thompson group F_k. Establish whether a comparable dynamical or cohomological obstruction exists for these families, or conversely whether such finite PL or projective actions can occur.

Background

The paper proves that for the family of forest‑skein groups built from the skein presentation (J), the derived T‑ and V‑type groups act on the circle but admit neither non‑trivial finite piecewise linear nor finite piecewise projective actions, via an obstruction that combines bounded cohomology and the Brin–Squier theorem excluding faithful finite PL actions of Z ∗ Z on R.

By contrast, the authors do not know if a similar obstruction applies to the other two large families they construct and paper: (G) (generalised Cleary FS categories, equating a monochromatic tree t(a) with a right‑vine ρ(b)) and (H) (Higman–Thompson‑type relations Y(i)(I ⊗ Y(j)) = Y(j)(Y(i) ⊗ I)). Determining whether analogous obstructions exist—or whether finite PL/projective actions actually do exist—for these families would clarify the landscape of dynamical limitations across forest‑skein groups.