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

Abstract

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,$\mathbb{C}$). 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.

Minor correction at the end of 6.2.1, and perfect figures!

References in corpus (2)

• The Amalgamated Product Structure of the Tame Automorphism Group in Dimension Three
• Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism

Cited by in corpus (9)

• Presqu'un immeuble pour le groupe des automorphismes modérés
• Tits type alternative for groups acting on toric affine varieties
• Degree growth for tame automorphisms of an affine quadric threefold
• On the acylindrical hyperbolicity of the tame automorphism group of ${\mathrm{SL}}_2(\mathbb{C})$
• Combinatorics of the tame automorphism group
• Acylindrical hyperbolicity of the three-dimensional tame automorphism group
• Automorphisms of certain affine complements in the projective space
• Tits-type alternative for certain groups acting on algebraic surfaces
• The Structure of the Three-Dimensional Special Linear Group over a Local Field