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

TL;DR

This paper investigates the automorphism groups of certain affine threefolds defined by word equations in SL(2,C)^2, establishing their invariants and Jordan property, thus contributing to algebraic geometry and group theory.

Contribution

It proves that the Makar-Limanov invariant equals the coordinate ring and that the automorphism group is Jordan for these specific varieties.

Findings

ML(S_{g}) equals the coordinate ring

Aut(S_{g}) is Jordan

Automorphism groups are explicitly characterized

Abstract

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