Elekes-Szab\'o for groups, and approximate subgroups in weak general position

TL;DR

This paper characterizes nilpotent complex algebraic groups using a weak general position version of the Elekes-Szabó condition and proves a Mordell-Lang result for generic subgroups.

Contribution

It introduces a new weak general position concept to characterize nilpotence and establishes a Mordell-Lang theorem for generic finitely generated subgroups.

Findings

Ekes-Szabó condition characterizes nilpotence in connected complex algebraic groups.

A Mordell-Lang result is proved for generic finitely generated subgroups.

The work links group structure with algebraic and combinatorial properties.

Abstract

We show that with a suitable weak notion of general position, the Elekes-Szab\'o condition on the group operation of a connected complex algebraic group characterises nilpotence of the group. Along the way, we prove a Mordell-Lang result for generic finitely generated subgroups of commutative complex algebraic groups.