A finitely presented infinite simple group of homeomorphisms of the circle

Abstract

We construct a finitely presented, infinite, simple group that acts by homeomorphisms on the circle, but does not admit a non-trivial action by $C^1$-diffeomorphisms on the circle. The group emerges as a group of piecewise projective homeomorphisms of $S^1=\\mathbf{R}\\cup \\{\\infty\\}$. However, we show that it does not admit a non-trivial action by piecewise linear homeomorphisms of the circle.