Example of non-linearizable quasi-cyclic subgroup of automorphism group of polynomial algebra

TL;DR

This paper constructs an example of a torsion subgroup in the automorphism group of a rank 2 polynomial algebra over complex numbers that is not conjugate to any subgroup of linear automorphisms, challenging previous assumptions.

Contribution

It provides the first known example of a non-linearizable torsion subgroup in the automorphism group of polynomial algebra of rank 2.

Findings

Finite subgroups are conjugate to linear automorphisms

Constructed abelian p-group is not conjugate to linear automorphisms

Challenges the generalization of linearization results

Abstract

It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear automorphisms. We prove that it is not true for an arbitrary torsion subgroup. We construct an example of abelian $p$-group of automorphism of polynomial algebra of rank 2 over the field of complex numbers, which is not conjugated with a subgroup of linear automorphisms.