Forest-skein groups IV: dynamics

We study forest-skein (FS) groups using dynamics. A simple Ore FS category produces three FS groups analogous to Richard Thompson's groups. Reconstruction theorems of McCleary and Rubin apply to these FS groups: each of them encodes a canonical rigid group action and thus carries powerful dynamical invariants. We then explicitly construct infinitely many isomorphism classes of finitely presented (of type $F_\infty$) infinite simple groups which act faithfully on the circle by (orientation-preserving) homeomorphisms, but admit no non-trivial finite piecewise linear actions nor finite piecewise projective actions. To the best of our knowledge these are the first examples witnessing these properties. We also show these groups fit into the finite germ extension framework of Belk, Hyde, and Matucci.