This paper is the first in a sequence on the first order theory of free products and further generalizations. In the first paper, we generalize the analysis of systems of equations over free and (torsion-free) hyperbolic groups, and analyze systems of equations over free products. To do that we introduce limit groups over the class of free products, and show that a finitely presented group has a canonical (finite) collection of maximal limit quotients. We further extend this finite collection and associate a Makanin–Razborov diagram over free products with every f.p. group. This MR diagram encodes all the quotients of a given f.p. group that are free products, all its homomorphisms into free products, and equivalently all the solutions to a given system of equations over a free product.
"Makanin–Razborov diagrams over free products." Illinois J. Math. 54 (1) 19 - 68, Spring 2010. https://doi.org/10.1215/ijm/1299679737