Geometric presentations for Thompsonʹs groups

Starting from the observation that Thompsonʹs groups F and V are the geometry groups respectively of associativity, and of associativity together with commutativity, we deduce new presentations of these groups. These presentations naturally lead to introducing a new subgroup image of V and a torsion free extension B• of image. We prove that image and B• are the geometry groups of associativity together with the law x(yz)=y(xz), and of associativity together with a twisted version of this law involving self-distributivity, respectively.