ML-invariant and automorphism groups of certain word varieties in SL(2,C)²

2025/12/13 by Tatiana Bandman, Bandman, Tatiana
Mathematics · #Advanced Algebra and Geometry #Holomorphic and Operator Theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2512.12126

Abstract

For a fixed element g∈ SL(2,C) and a word w=[xⁿ,y^m] we consider the automorphism group Aut(S_g) of the affine threefold S_g=\(x,y)∈ SL(2,C)² | w(x,y)=g\. We prove that Makar-Limanov invariant ML(S_g)=O(S_g) and Aut(S_g) is Jordan.

Citations

• Birational properties of word varieties
• Jordan property for automorphism groups of compact varieties
• Jordan properties of automorphism groups of certain open algebraic varieties
• Surjectivity of certain word maps on PSL(2,C) and SL(2,C)
• Jordan groups and algebraic surfaces
• Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics
• Jordan property for Cremona groups
• Criteria for equidistribution of solutions of word equations on SL(2)
• Theta groups and products of abelian and rational varieties
• A Minkowski-style bound for the order of the finite subgroups of the Cremona group of rank 2 over an arbitrary field
• Simultaneous similarity of matrices

Related

• Automorphism groups of affine varieties and their Lie algebras
• Automorphisms of wonderful varieties
• Birational properties of word varieties
• Makar-Limanov invariants of nonnormal affine toric varieties
• On automorphisms of Danielewski surfaces