Affine surfaces with a huge group of automorphisms

TL;DR

This paper constructs a family of rational affine surfaces with exceptionally large automorphism groups, exhibiting complex subgroup structures and uncountably generated quotients, revealing new insights into surface symmetries.

Contribution

It introduces a novel family of affine surfaces with automorphism groups exhibiting uncountably generated quotients and complex subgroup structures.

Findings

The automorphism group contains a normal subgroup generated by all algebraic subgroups.

The quotient of the automorphism group by this subgroup contains a free group over an uncountable set.

The structure of automorphism groups on these surfaces is highly intricate and large.

Abstract

We describe a family of rational affine surfaces S with huge groups of automorphisms in the following sense: the normal subgroup of Aut(S) generated by all its algebraic subgroups is not generated by any countable family of such subgroups, and the quotient of Aut(S) by this subgroup contains a free group over an uncountable set of generators.