On the elementary theory of graph products of groups

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 that generalizes a range of natural "non-freeness conditions" such as the satisfaction of a group law, having a non -trivial 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 and describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results for the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent, or classical linear groups.