Geometric presentations for Thompson's groups

Starting from the observation that Thompson's 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 S
- of V and a torsion free extension B
- of S
- . We prove that S
- 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.