Generalizing a question of Gromov

Gromov asked, "What does a typical group look like? He suggested a way of describing typical behavior in terms of limiting density. Based on a remark of Fine, I conjectured that for groups on $n\geq 2$ generators, and presentations with a single relator, an elementary first order sentence has limiting density $1$ iff it is true in the non-Abelian free groups. There are partial positive results due to Coulon, Ho, and Logan, and to Karlampovich and Sklinos, but the full conjecture remains open. I will describe joint work with Johanna Franklin and Meng-Che (Turbo) Ho, in which we generalize Gromov's question to other algebraic varieties (in the sense of universal algebra). We have results and examples illustrating different possible behaviors.