C1 actions and distortion for the newly constructed FS groups

Determine whether the simple forest‑skein groups constructed in this paper, in particular those arising from the skein presentation (J), admit any C1 actions on any 1‑manifold and whether they contain a distorted element. This problem asks to decide if these groups share Lodha’s two properties: absence of C1 actions on 1‑manifolds and presence of a distorted element.

Background

The authors show that the T‑ and V‑type simple groups produced from the (J) skein presentation act on the circle but admit no non‑trivial finite PL or finite piecewise projective actions. They compare these groups to Lodha’s finitely presented simple group acting piecewise projectively on S1, which is known to have two notable properties: it does not admit any C1 actions on any 1‑manifold and it contains a distorted element.

It remains to be determined whether the forest‑skein groups constructed here share these two further properties. Resolving this would position the new examples within the broader taxonomy of simple groups of homeomorphisms of 1‑manifolds.