We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{ SL}(2,\mathbb{ Z})$, the torus and a special map of order $5$, as it was conjectured by A. Usnich. Then we consider a special subgroup $H$, of finite type, defined over any field which admits a surjective morphism to the Thompson group of piecewise linear automorphisms of $\mathbb{ Z}^2$. We prove that the presentation for this group conjectured by Usnich is correct.
Categories: math.AG math.DG