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

Cinzia Bisi[1]; Jean-Philippe Furter[2]; Stéphane Lamy[3]

  • [1] Dipartimento Matematica ed Informatica, Universitá di Ferrara Via Machiavelli n.35, 44121 Ferrara, Italy
  • [2] Laboratoire MIA, Université de La Rochelle Avenue Michel Crépeau, 17000 La Rochelle, France
  • [3] Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 9, France

Journal de l’École polytechnique — Mathématiques (2014)

  • Volume: 1, page 161-223
  • ISSN: 2270-518X

Abstract

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We study the group Tame ( SL 2 ) of tame automorphisms of a smooth affine 3 -dimensional quadric, which we can view as the underlying variety of SL 2 ( ℂ ) . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT ( 0 ) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame ( SL 2 ) is linearizable, and that Tame ( SL 2 ) satisfies the Tits alternative.

How to cite

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Bisi, Cinzia, Furter, Jean-Philippe, and Lamy, Stéphane. "The tame automorphism group of an affine quadric threefold acting on a square complex." Journal de l’École polytechnique — Mathématiques 1 (2014): 161-223. <http://eudml.org/doc/275564>.

@article{Bisi2014,
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.},
affiliation = {Dipartimento Matematica ed Informatica, Universitá di Ferrara Via Machiavelli n.35, 44121 Ferrara, Italy; Laboratoire MIA, Université de La Rochelle Avenue Michel Crépeau, 17000 La Rochelle, France; Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 9, France},
author = {Bisi, Cinzia, Furter, Jean-Philippe, Lamy, Stéphane},
journal = {Journal de l’École polytechnique — Mathématiques},
keywords = {Automorphism group; affine quadric; cube complex; Tits alternative; automorphism group},
language = {eng},
pages = {161-223},
publisher = {École polytechnique},
title = {The tame automorphism group of an affine quadric threefold acting on a square complex},
url = {http://eudml.org/doc/275564},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Bisi, Cinzia
AU - Furter, Jean-Philippe
AU - Lamy, Stéphane
TI - The tame automorphism group of an affine quadric threefold acting on a square complex
JO - Journal de l’École polytechnique — Mathématiques
PY - 2014
PB - École polytechnique
VL - 1
SP - 161
EP - 223
AB - 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.
LA - eng
KW - Automorphism group; affine quadric; cube complex; Tits alternative; automorphism group
UR - http://eudml.org/doc/275564
ER -

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