Degree of satisfiability of some special equations

TL;DR

This paper investigates the probability that certain algebraic equations hold in groups, showing that some equations are either always satisfied or rarely satisfied, and introduces criteria to identify equations with positive satisfiability in finite index subgroups.

Contribution

It establishes bounds on the degree of satisfiability for specific equations and develops criteria to determine when equations hold in finite index subgroups.

Findings

Degree of satisfiability for certain equations is either 1 or bounded away from 1 by a positive constant.

Introduces criteria to identify equations with positive satisfiability in finite index subgroups.

Shows that some equations do not have the property of holding in a finite index subgroup if they have positive degree of satisfiability.

Abstract

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.