Finitely presented simple torsion-free groups in the landscape of Richard Thompson

We construct the first (to our knowledge) examples of finitely presented (infinite) simple groups of homeomorphisms of R. Our examples also satisfy the stronger finiteness property type $\mathbf{F}_{\infty}$: they are fundamental groups of connected aspherical CW complexes with finitely many cells in each dimension. To our knowledge, ours is the only family of finitely presented simple torsion-free groups that has been constructed since the Burger-Mozes construction from [8]. Besides admitting faithful actions on the line by homeomorphisms, the groups in our family satisfy the following fundamentally new features. They have infinite geometric (and cohomological) dimension, and they admit a nontrivial homogeneous quasimorphism (and hence have infinite commutator width).