Gromov asked what a typical group looks like, and he suggested a way to make the question precise in terms of limiting density. The typical finitely presented group is known to share some important properties with the non-abelian free groups. Knight conjectured that the typical group satisfies a zero-one law and has the same first-order theory as the free group. Kharlampovich and Sklinos verified this conjecture for one-quantifier sentences.

We generalize Gromov's notion and Knight's question to structures in an arbitrary algebraic variety (in the sense of universal algebra). We give examples illustrating different behaviors of the limiting density. Based on the examples, we identify sufficient conditions for the elementary first-order theory of the free structure to match that of the typical structure; i.e., a sentence is true in the free structure if and only if it has limiting density 1.

This is joint work with Johanna Franklin and Julia Knight.