2020/02/05 by Zoltan A. Kocsis, Kocsis, Zoltan A.
Mathematics · #20D60 (Primary) 20P05 (Secondary) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2002.01773
openalex publication_date 2020/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (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 xy2=y2x, xy3=y3x and xy=yx-1 is either 1, or no larger than 1-ε for some positive constant ε. 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 xy2=y2x do not have this property.