Free structures and limiting density

Gromov asked what a typical (finitely presented) group looks like, and he suggested a way to make the question precise in terms of limiting density. The typical finitely generated group is known to share some important properties with the non-Abelian free groups. The third author conjectured that for n≥2 and group presentations with a single relator, sentences true in the non-Abelian free groups have density 1. We state Gromov’s question more generally, for structures in an arbitrary algebraic variety (in the sense of universal algebra), with presentations of a specific form, and focusing on elementary (i.e., finitary) first order sentences. We give examples illustrating different behaviors. In the first, the sentences true in the typical structure are just those true in the free structure. In the second, each sentence has limiting density 0 or 1, but the theory of the typical structure is not that of the free structure. In the third, there is a sentence with limiting density strictly between 0 and 1. In the fourth, the limiting density lies strictly between 0 and 1 for some sentences, and does not exist for some other sentences. Generalizing the first example, we consider commutative generalized bijective varieties in a language consisting of finitely many unary function symbols. We show that for presentations with a single generator and a single identity, if the free structure F is infinite, then the theory of the typical structure matches that of F. This also holds for presentations with m generators and one identity.