Commuting automorphisms and characterization of an affine space

Our base field is an uncountable algebraically closed field (of any characteristic). We show that an affine space is determined by the abstract group structure of its regular automorphism group in the category of connected affine varieties. To prove this we use tools from birational geometry and study commutative subgroups of the group of automorphisms of affine varieties.
(This is joint work with Serge Cantat and Junyi Xie).