Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups

TL;DR

This paper develops algebraic geometry tools for partially commutative groups, establishing criteria for group properties, normal forms for formulas, and proving the decidability of their positive theory.

Contribution

It introduces a criterion for a partially commutative group to be a domain and proves quantifier elimination for the positive theory of non-abelian indecomposable groups.

Findings

Criterion for a group to be a domain

Normal forms for quantifier-free formulas

Decidability of the positive theory

Abstract

In this version small mistakes are corrected and the exposition is changed as suggested by the referee (to appear in Canadian Journal of Mathematics). The first main result of the paper is a criterion for a partially commutative group $\GG$ to be a domain. It allows us to reduce the study of algebraic sets over $\GG$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\GG$ (of a coordinate group over $\GG$) to the elementary theories of the direct factors of $\GG$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\HH$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\HH$ has quantifier elimination and that arbitrary first-order formulas lift from $\HH$ to $\HH\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.