The tame automorphism group of an affine quadric threefold acting on a square complex

[Action du groupe modéré d’une quadrique affine de dimension $3$ sur un complexe carré]

Bisi, Cinzia ; Furter, Jean-Philippe ; Lamy, Stéphane

Journal de l'École polytechnique - Mathématiques, Tome 1 (2014), p. 161-223 / Harvested from Numdam

Résumé

Nous étudions le groupe $\mathrm{Tame}(\mathrm{SL}_2)$ des automorphismes modérés d’une quadrique affine lisse de dimension $3$, que l’on peut choisir comme étant la variété sous-jacente à $\mathrm{SL}_2(\mathbb{C})$. Nous construisons un complexe carré sur lequel ce groupe agit naturellement de façon cocompacte, et nous montrons que ce complexe est $\mathrm{CAT}(0)$ et hyperbolique. Nous proposons ensuite deux applications de cette construction : nous montrons que tout sous-groupe fini de $\mathrm{Tame}(\mathrm{SL}_2)$ est linéarisable, et que $\mathrm{Tame}(\mathrm{SL}_2)$ satisfait l’alternative de Tits.

Abstract

We study the group $\mathrm{Tame}(\mathrm{SL}_2)$ of tame automorphisms of a smooth affine $3$-dimensional quadric, which we can view as the underlying variety of $\mathrm{SL}_2(\mathbb{C})$. We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $\mathrm{CAT}(0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $\mathrm{Tame}(\mathrm{SL}_2)$ is linearizable, and that $\mathrm{Tame}(\mathrm{SL}_2)$ satisfies the Tits alternative.

Détail

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/jep.8
Classification:  14J50,  14R20,  20F65
Mots clés: Groupe d’automorphismes, quadrique affine, complexe cubique, alternative de Tits

Bibliographie

[AFK + 13] Arzhantsev, I.; Flenner, H.; Kaliman, S.; Kutzschebauch, F.; Zaidenberg, M. Flexible varieties and automorphism groups, Duke Math. J., Tome 162 (2013) no. 4, pp. 767-823 | MR 3039680 | Zbl pre06157711

[Ale95] Alev, J. A note on Nagata’s automorphism, Automorphisms of affine spaces (Curaçao, 1994), Kluwer Acad. Publ., Dordrecht (1995), pp. 215-221 | MR 1352702 | Zbl 0831.14004

[AOS12] Ardila, F.; Owen, M.; Sullivant, S. Geodesics in $\rm CAT(0)$ cubical complexes, Adv. in Appl. Math., Tome 48 (2012) no. 1, pp. 142-163 | MR 2845512 | Zbl 1275.05055

[BH99] Bridson, M. R.; Haefliger, A. Metric spaces of non-positive curvature, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome 319 (1999) | MR 1744486 | Zbl 0988.53001

[BŚ99] Ballmann, W.; Świątkowski, J. On groups acting on nonpositively curved cubical complexes, Enseign. Math. (2), Tome 45 (1999) no. 1-2, pp. 51-81 | MR 1703363 | Zbl 0989.20029

[Can11] Cantat, S. Sur les groupes de transformations birationnelles des surfaces, Ann. of Math. (2), Tome 174 (2011) no. 1, pp. 299-340 | MR 2811600 | Zbl 1233.14011

[CL13] Cantat, S.; Lamy, S. Normal subgroups in the Cremona group, Acta Math., Tome 210 (2013) no. 1, pp. 31-94 (With an appendix by Yves de Cornulier) | MR 3037611 | Zbl 1278.14017

[Din12] Dinh, T.-C. Tits alternative for automorphism groups of compact Kähler manifolds, Acta Math. Vietnamatica, Tome 37 (2012) no. 4, pp. 513-529 | MR 3058661 | Zbl 1271.14056

[dlH83] De La Harpe, P. Free groups in linear groups, Enseign. Math. (2), Tome 29 (1983) no. 1-2, pp. 129-144 | MR 702736 | Zbl 0517.20024

[Fru73] Frumkin, M. A. A filtration in the three-dimensional Cremona group, Mat. Sb. (N.S.), Tome 90(132) (1973), p. 196-213, 325 | MR 327769 | Zbl 0254.14006

[Fur83] Furushima, M. Finite groups of polynomial automorphisms in ${\bf C}^{n}$, Tohoku Math. J. (2), Tome 35 (1983) no. 3, pp. 415-424 | MR 711357 | Zbl 0567.32010

[GD77] Gizatullin, M. H.; Danilov, V. I. Automorphisms of affine surfaces. II, Izv. Akad. Nauk SSSR Ser. Mat., Tome 41 (1977) no. 1, pp. 54-103 | MR 437545 | Zbl 0357.14003

[Kam79] Kambayashi, T. Automorphism group of a polynomial ring and algebraic group action on an affine space, J. Algebra, Tome 60 (1979) no. 2, pp. 439-451 | MR 549939 | Zbl 0429.14017

[Kur10] Kuroda, S. Shestakov-Umirbaev reductions and Nagata’s conjecture on a polynomial automorphism, Tohoku Math. J. (2), Tome 62 (2010) no. 1, pp. 75-115 | MR 2654304 | Zbl 1210.14072

[Lam01] Lamy, S. L’alternative de Tits pour ${\rm Aut}[{\mathbb{C}}^2]$, J. Algebra, Tome 239 (2001) no. 2, pp. 413-437 | MR 1832900 | Zbl 1040.37031

[Lam13] Lamy, S. On the genus of birational maps between 3-folds (2013) (arXiv:1305.2482)

[LV13] Lamy, S.; Vénéreau, S. The tame and the wild automorphisms of an affine quadric threefold., J. Math. Soc. Japan, Tome 65 (2013) no. 1, pp. 299-320 | MR 3034406 | Zbl pre06152435

[Pan99] Pan, I. Une remarque sur la génération du groupe de Cremona, Bol. Soc. Brasil. Mat. (N.S.), Tome 30 (1999) no. 1, pp. 95-98 | MR 1686984 | Zbl 0972.14006

[Pap95] Papasoglu, P. Strongly geodesically automatic groups are hyperbolic, Invent. Math., Tome 121 (1995) no. 2, pp. 323-334 | MR 1346209 | Zbl 0834.20040

[PV91] Pays, I.; Valette, A. Sous-groupes libres dans les groupes d’automorphismes d’arbres, Enseign. Math. (2), Tome 37 (1991) no. 1-2, pp. 151-174 | MR 1115748 | Zbl 0744.20024

[Ser77a] Serre, J.-P. Arbres, amalgames, ${\rm SL}_{2}$, Société Mathématique de France, Paris, Astérisque, Tome 46 (1977), pp. 189 p. | MR 476875 | Zbl 0369.20013

[Ser77b] Serre, J.-P. Cours d’arithmétique, Presses Universitaires de France, Paris (1977) | MR 498338 | Zbl 0376.12001

[Wis12] Wise, D. T. From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, American Mathematical Society, Providence, RI, CBMS Regional Conference Series in Mathematics, Tome 117 (2012) | MR 2986461 | Zbl 1278.20055

[Wri13] Wright, D. The Amalgamated Product Structure of the Tame Automorphism Group in Dimension Three (2013) (arXiv:1310.8325)

[Wri92] Wright, D. Two-dimensional Cremona groups acting on simplicial complexes, Trans. Amer. Math. Soc., Tome 331 (1992) no. 1, pp. 281-300 | MR 1038019 | Zbl 0767.14006