Abstract

In a finite group G one can ask what the probability is that two elements chosen independently uniformly at random commute. It is clear that if G has an abelian subgroup of bounded index then this probability should be bounded from below. A beautiful theorem of Peter Neumann from the 1980s shows that the converse is also true in a certain precise sense.

Antolin, Martino and Ventura consider the same question for infinite groups, choosing the two random elements with respect to a certain limit of finite probability measures. They conjecture that for any “reasonable” sequence of measures Neumann’s result should still hold. They also prove some special cases of this conjecture.

In this talk I will show that the Antolin-Martino-Ventura conjecture holds with effective quantitative bounds if we take the sequence of measures defined by the successive steps of a simple random walk, or the uniform measures on a Folner sequence. I will also present a concrete interpretation of the word “reasonable” that is sufficient to force a sequence of measures to obey the conjecture. If I have time I will present an application to conjugacy growth.