Charles Cox

Degree of commutativity of infinite groups

The degree of commutativity of a finite group is the probability that any two elements commute i.e. the proportion of pairs a, b from G^2 such that ab=ba. Therefore a group is abelian if and only if this number is 1. When Gustafson introduced this idea he showed a 'gap' result: there are no groups with degree of commutativity strictly between 5/8 and 1.

This concept was recently generalised to infinite finitely generated groups by Yago Antolin, Armando Martino, and Enric Ventura. I will explain their approach, give a gentle introduction to the relevant ideas of growth and conjugacy growth, and explain the main conjecture, progress, and other ideas.