Finitely presented simple groups with no piecewise projective actions

Abstract

We construct an explicit infinite family of pairwise non-isomorphic infinite simple groups of type ${\mathrm{F}}_{\infty }$ (in particular, they are finitely presented) that act faithfully on the circle by orientation-preserving homeomorphisms, but that admit no 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.

32 pages, 18 figures, this article is a shorter self-contained version of arXiv:2411.12569 focused on witnessing corollary F, to appear in J. Lond. Math. Soc