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A geometric proof of a theorem of Jung. (Une preuve géométrique du théorème de Jung.) (French) Zbl 1044.14035

Let \(\text{Aut}({\mathbb C}^2)\) be the group of polynomial automorphisms of \({\mathbb C}^2\). Let \(A \subset \text{Aut}({\mathbb C}^2)\) be the subgroup of affine automorphisms, i.e. \[ A = \{(x,y) \rightarrow (a_1x+b_1y+c_1,a_2x+b_2y+c_2) \mid a_i,b_i,c_i \in {\mathbb C}, a_1b_2-a_2b_1 \neq 0\}, \] and \(E \subset \text{Aut}({\mathbb C}^2)\) be the subgroup of elementary automorphisms, i.e. \[ E = \{(x,y) \rightarrow (ax+p(y),by+c) | a,b \in {\mathbb C}^*, p \in {\mathbb C}[y]\}. \] H. E. W. Jung proved that \(\text{Aut}({\mathbb C}^2)\) is generated by \(A\) and \(E\) [J. Reine Angew. Math. 184, 161–174 (1942; Zbl 0027.08503)]. A new geometric proof of this theorem is given.

MSC:

14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
14L40 Other algebraic groups (geometric aspects)
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