Finitely presented simple groups with no piecewise projective actions

TL;DR

This paper constructs an infinite family of finitely presented simple groups that act faithfully on the circle but do not admit non-trivial piecewise affine or projective actions, revealing new properties of such groups.

Contribution

It introduces explicit examples of simple groups with specific action restrictions, combining ideas from Thompson groups and planar algebras.

Findings

Constructed infinite simple groups of type F_infinity

Groups act faithfully on the circle by orientation-preserving homeomorphisms

Groups admit no non-trivial piecewise affine or 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.