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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. This is the first such example. The group emerges as a group of piecewise projective homeomorphisms ofS  1   =  R  ∪   {  ∞  }    . We also show that it does not admit a non‐trivial action by piecewise linear homeomorphisms of the circle. Another interesting and new feature of this example is that it produces a non‐amenable orbit equivalence relation with respect to the Lebesgue measure.

A finitely presented infinite simple group of homeomorphisms of the circle