Finitely presented simple groups with no piecewise projective actions

We construct an explicit infinite family of pairwise non‐isomorphic infinite simple groups of type (in particular, they are finitely presented) that act faithfully on the circle by orientation‐preserving homeomorphisms, but that admit neither non‐trivial piecewise affine nor piecewise projective actions on the projective line. Our examples are certain forest‐skein groups which, informally, are a mixture of Richard Thompson's groups with Vaughan Jones' planar algebras.