Non‑amenability of G_t

Establish whether the F‑type forest‑skein groups G_t constructed from the skein presentation (G) are non‑amenable. Concretely, determine amenability of G_t, where t is a monochromatic tree and (G) is FS a,b|t(a) = ρ(b) with ρ a right‑vine.

Background

In the discussion comparing properties of the various forest‑skein groups, the authors note that the G_t groups (from the (G) family) do not contain non‑abelian free subgroups (specifically Z ∗ Z), but they do not know whether these groups are non‑amenable.

Amenability is a fundamental property with implications for dynamics, geometry, and analysis of groups. Determining amenability of G_t would help situate these groups among known classes of Thompson‑like groups and clarify how the absence of free subgroups interacts with large‑scale dynamics in the forest‑skein framework.