A finitely presented infinite simple group of homeomorphisms of the circle

TL;DR

This paper constructs a finitely presented, infinite simple group acting on the circle via homeomorphisms, which cannot be realized through smoother or piecewise linear actions, and exhibits non-amenable orbit relations.

Contribution

It introduces a new finitely presented simple group with unique action properties and explores its non-amenable orbit equivalence relation.

Findings

The group acts by homeomorphisms but not by $C^1$-diffeomorphisms.

It does not admit a non-trivial piecewise linear action.

The group produces a non-amenable orbit equivalence relation.

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. Another interesting and new feature of this example is that it produces a non amenable orbit equivalence relation with respect to the Lebesgue measure.