On the elementary theory of graph products of groups

TL;DR

This paper investigates how the elementary theory of graph products of groups encodes the structure of the underlying graph and vertex groups, leading to rigidity and classification results.

Contribution

It introduces conditions under which the elementary theory determines the core graph and vertex groups, extending to right-angled Artin groups and their elementary classification.

Findings

The elementary theory can recover the core graph and vertex groups under natural conditions.

The core of the defining graph is an invariant of the elementary theory of RAAGs.

Results include elementary classification and rigidity for specific vertex group types.

Abstract

In this paper we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we require the vertex groups to satisfy a non-generic almost positive sentence, a condition which generalizes a range of natural ``non-freeness conditions" such as the satisfaction of a group law, having nontrivial center or being boundedly simple. As a corollary, we determine an invariant of the elementary theory of a right-angled Artin group, the core of the defining graph, which we conjecture to determine the elementary class of the RAAG. We further combine our results with the results of Sela on free products of groups to describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results on the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent or classical linear groups.