Inflations of geometric grid classes of permutations

Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than $\kappa\approx2.20557$ (a specific algebraic integer at which infinite antichains begin to appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are partially well-ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is partially well-ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than $\kappa$ has a rational generating function. This bound is tight as there are permutation classes with growth rate $\kappa$ which have nonrational generating functions.