Degree of satisfiability of some special equations

A well-known theorem of Gustafson states that in a non-Abelian group the degree of satisfiability of $xy=yx$, i.e. the probability that two uniformly randomly chosen group elements $x,y$ obey the equation $xy=yx$, is no larger than $\frac{5}{8}$. The seminal work of Antolin, Martino and Ventura (arXiv:1511.07269) on generalizing the degree of satisfiability to finitely generated groups led to renewed interest in Gustafson-style properties of other equations. Positive results have recently been obtained for the 2-Engel and metabelian identities (arXiv:1809.02997). Here we show that the degree of satisfiability of the equations $xy^2=y^2x$, $xy^3=y^3x$ and $xy=yx^{-1}$ is either 1, or no larger than $1-\varepsilon$ for some positive constant $\varepsilon$. Using the Antolin-Martino-Ventura formalism, we introduce criteria to identify which equations hold in a finite index subgroup precisely if they have positive degree of satisfiability. We deduce that the equations $xy=yx^{-1}$ and $xy^2=y^2x$ do not have this property.

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