On the group of polynomial antomorphisms of the affine planes. (Sur le groupe des automorphismes polynomiaux du plan affine.) (English) Zbl 1096.14046

The main result of the paper under review is that, over a uncountable field \(k\) of characteristic 0, the automorphism group of the group \(\text{Aut}(k^2)\) of polynomial automorphisms of the affine plane is generated by the inner automorphisms of \(\text{Aut}(k^2)\) and the action of automorphisms of the field: If \(\varphi\) is an automorphism of the group \(\text{Aut}(k^2)\), then there exists a polynomial automorphism \(\psi\) and an automorphism \(\tau\) of the field \(k\) such that \(\varphi(f)=\tau(\psi f\psi^{-1})\) for all \(f\in \text{Aut}(k^2)\).
As an application, the author shows that an automorphism \(\varphi\) of \(\text{Aut}(k^2)\) is inner if and only if it preserves the determinant of the Jacobian matrix of the automorphisms of \(k^2\). Another application gives that the automorphisms of the semigroup \(\text{End}(k^2)\) of polynomial endomorphisms of \(k^2\) are compositions of inner automorphisms and automorphisms of the field.


14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
13B25 Polynomials over commutative rings
14E07 Birational automorphisms, Cremona group and generalizations
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
Full Text: DOI


[1] Ahern, P.; Rudin, W., Periodic automorphisms of \(\mathbf{C}^n\), Indiana univ. math. J., 44, 1, 287-303, (1995) · Zbl 0838.32010
[2] Akhiezer, D.N., Lie group actions in complex analysis, Aspects math., vol. 27, (1995), Vieweg Braunschweig · Zbl 0845.22001
[3] Banyaga, A., On isomorphic classical diffeomorphism groups I, Proc. amer. math. soc., 98, 1, 113-118, (1986) · Zbl 0602.53027
[4] Banyaga, A., The structure of classical diffeomorphism groups, Math. appl., vol. 400, (1997), Kluwer Acad. Publ. Dordrecht · Zbl 0874.58005
[5] Buzzard, G.T.; Merenkov, S., Maps conjugating holomorphic maps in \(\mathbb{C}^n\), Indiana univ. math. J., 52, 5, 1135-1146, (2003) · Zbl 1084.32502
[6] Cerveau, D.; Ghys, É.; Sibong, N.; Yoccoz, J.C., Dynamique et Géométrie complexes, Panor. synthèses, vol. 8, (1999), Soc. Math. France Paris
[7] Dieudonné, J., La Géométrie des groupes classiques, (1963), Springer Berlin · Zbl 0111.03102
[8] Filipkiewicz, R.P., Isomorphisms between diffeomorphism groups, Ergodic theory dynam. systems, 2, 2, 159-171, (1982) · Zbl 0521.58016
[9] Friedland, S.; Milnor, J., Dynamical properties of plane polynomial automorphisms, Ergodic theory dynam. systems, 9, 1, 67-99, (1989) · Zbl 0651.58027
[10] Lamy, S., L’alternative de Tits pour \(\operatorname{Aut} [\mathbb{C}^2]\), J. algebra, 239, 2, 413-437, (2001) · Zbl 1040.37031
[11] Lamy, S., Une preuve géométrique du théorème de jung, Enseign. math. (2), 48, 3-4, 291-315, (2002) · Zbl 1044.14035
[12] Smillie, J., Dynamics in two complex dimensions, (), 373-382 · Zbl 1136.37341
[13] van der Kulk, W., On polynomial rings in two variables, Nieuw arch. wisk. (3), 1, 33-41, (1953) · Zbl 0050.26002
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