On the maximality of the triangular subgroup. (Sur la maximalité du sous-groupe triangulaire.) (English. French summary) Zbl 1447.14008
The group \(\text{Aut}(\mathbb A_{\mathbb C}^n)\) of polynomial automorphisms of the affine complex space carries a natural structure of infinite dimensional algebraic group. The authors are interested in maximality properties of the subgroup \(B_n\) of triangular automorphism, which is a closed solvable subgroup. Their main result states that for any \(n \ge 2\), \(B_n\) is maximal among all solvable subgroups of \(\text{Aut}(\mathbb A_{\mathbb C}^n)\). Moreover in the case \(n = 2\), they show that \(B_2\) is maximal among closed subgroups of \(\text{Aut}(\mathbb A_{\mathbb C}^2)\), but is not maximal among all subgroups.
Reviewer: Stéphane Lamy (Toulouse)
MSC:
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
| 20G99 | Linear algebraic groups and related topics |
References:
| [1] | Hyman Bass, “A nontriangular action of \(\mathbb{G}_a\) on \(\mathbb{A}^3\)”, J. Pure Appl. Algebra33 (1984) no. 1, p. 1-5 · Zbl 0555.14019 · doi:10.1016/0022-4049(84)90019-7 |
| [2] | Hyman Bass, Edwin H. Connell & David Wright, “The Jacobian conjecture: reduction of degree and formal expansion of the inverse”, Bull. Am. Math. Soc.7 (1982) no. 2, p. 287-330 · Zbl 0539.13012 · doi:10.1090/S0273-0979-1982-15032-7 |
| [3] | Arnaud Beauville, Finite subgroups of \(\operatorname{ PGL }_2(K)\), Vector bundles and complex geometry, Contemporary Mathematics 522, American Mathematical Society, 2010, p. 23-29 · Zbl 1218.20030 · doi:10.1090/conm/522/10289 |
| [4] | Yuri Berest, Alimjon Eshmatov & Farkhod Eshmatov, “Dixmier groups and Borel subgroups”, Adv. Math.286 (2016), p. 387-429 · Zbl 1400.20048 · doi:10.1016/j.aim.2015.09.012 |
| [5] | Éric Edo, “Closed subgroups of the polynomial automorphism group containing the affine subgroup”, to appear in \(Transform. Groups\) · Zbl 1388.13024 · doi:10.1007/s00031-016-9412-7 |
| [6] | Éric Edo & Pierre-Marie Poloni, “On the closure of the tame automorphism group of affine three-space”, Int. Math. Res. Not. (2015) no. 19, p. 9736-9750 · Zbl 1338.14057 · doi:10.1093/imrn/rnu237 |
| [7] | Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics 190, Birkhäuser, 2000 · Zbl 0962.14037 · doi:10.1007/978-3-0348-8440-2 |
| [8] | Shmuel Friedland & John Milnor, “Dynamical properties of plane polynomial automorphisms”, Ergodic Theory Dyn. Syst.9 (1989) no. 1, p. 67-99 · Zbl 0651.58027 · doi:10.1017/S014338570000482X |
| [9] | Jean-Philippe Furter, “On the length of polynomial automorphisms of the affine plane”, Math. Ann.322 (2002) no. 2, p. 401-411 · Zbl 0996.14031 · doi:10.1007/s002080100276 |
| [10] | Jean-Philippe Furter & Stéphane Lamy, “Normal subgroup generated by a plane polynomial automorphism”, Transform. Groups15 (2010) no. 3, p. 577-610 · Zbl 1226.14080 · doi:10.1007/s00031-010-9095-4 |
| [11] | James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics 21, Springer, 1975 · Zbl 0325.20039 |
| [12] | Heinrich W. E. Jung, “Über ganze birationale Transformationen der Ebene”, J. Reine Angew. Math.184 (1942), p. 161-174 · Zbl 0027.08503 · doi:10.1515/crll.1942.184.161 |
| [13] | Shulim Kaliman, Isotopic embeddings of affine algebraic varieties into \(\Bbb C^n\), The Madison Symposium on Complex Analysis (Madison, WI, 1991), Contemporary Mathematics 137, American Mathematical Society, 1992, p. 291-295 · Zbl 0768.14005 · doi:10.1090/conm/137/1190990 |
| [14] | W. van der Kulk, “On polynomial rings in two variables”, Nieuw Arch. Wiskunde1 (1953), p. 33-41 · Zbl 0050.26002 |
| [15] | Stéphane Lamy, “L’alternative de Tits pour \(####\) |
| [16] | A. L. Malʼcev, “On certain classes of infinite solvable groups”, Amer. Math. Soc. Transl.2 (1956), p. 1-21 · Zbl 0070.25404 |
| [17] | Mitchael Martelo & Javier Ribón, “Derived length of solvable groups of local diffeomorphisms”, Math. Ann.358 (2014) no. 3-4, p. 701-728 · Zbl 1328.37028 · doi:10.1007/s00208-013-0975-5 |
| [18] | Hirosi Nagao, “On \(####\) |
| [19] | Masayoshi Nagata, On automorphism group of \(####\) · Zbl 0306.14001 |
| [20] | Mike Frederick Newman, “The soluble length of soluble linear groups”, Math. Z.126 (1972), p. 59-70 · Zbl 0223.20055 · doi:10.1007/BF01580356 |
| [21] | Vladimir L. Popov, On actions of \(\mathbb{G}_a\) on \(\mathbb{A}^n\), Algebraic groups (Utrecht, 1986), Lecture Notes in Mathematics 1271, Springer, 1987, p. 237-242 · Zbl 0685.14027 · doi:10.1007/BFb0079241 |
| [22] | Vladimir L. Popov, “On infinite dimensional algebraic transformation groups”, Transform. Groups19 (2014) no. 2, p. 549-568 · Zbl 1309.14036 · doi:10.1007/s00031-014-9264-y |
| [23] | Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer, 2003, Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation |
| [24] | Igor R. Shafarevich, “On some infinite-dimensional groups”, Rend. Mat. Appl.25 (1966) no. 1-2, p. 208-212 · Zbl 0149.39003 |
| [25] | Igor R. Shafarevich, “On some infinite-dimensional groups. II”, Izv. Akad. Nauk SSSR Ser. Mat.45 (1981) no. 1, p. 214-226, 240 · Zbl 0475.14036 |
| [26] | Bertram A. F. Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse der Matematik und ihrer Grenzgebiete 76, Springer, 1973 · Zbl 0261.20038 |
| [27] | Hans Zassenhaus, “Beweis eines satzes über diskrete gruppen”, Abh. Math. Sem. Univ. Hamburg12 (1937) no. 1, p. 289-312 © Annales de L’Institut Fourier - ISSN (électronique) : 1777-5310 · Zbl 0023.01403 · doi:10.1007/BF02948950 |
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