Julia Knight (University of Notre Dame)

Freeness and typical behavior for algebraic structures

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The talk is on joint work with Johanna Franklin and Turbo Ho. Gromov asked “What is a typical group?” He was thinking of finitely presented groups. He proposed an approach involving limiting density. In 2013, I conjectured that for elementary first order sentences $\varphi$, and for group presentations with $n$ generators ($n\geq 2$) and a single relator, the limiting density for groups satisfying $\varphi$ always exists, with value $0$ or $1$, and the value is $1$ iff $\varphi$ is true in the non-Abelian free groups. The conjecture is still open, but there are positive partial results by Kharlampovich and Sklinos, and by Coulon, Ho, and Logan. We ask Gromov's question about structures in other equational classes, or algebraic varieties in the sense of universal algebra. We give examples illustrating different possible behaviors. Focusing on languages with just finitely many unary function symbols, we prove a result with conditions sufficient to guarantee that the analogue of the conjecture holds. The proof uses a version of Gaifman's Locality Theorem, plus ideas from random group theory and probability.