Symplectic birational transformations of the plane

Abstract

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.

14 pages

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