Means on groups and degrees of commutativityGeometric Group Theory Oberseminar

by Dr Motiejus Valiunas (University of Wrocław)

Wednesday Nov 5, 2025, 11:00 AM → 12:00 PM Europe/Berlin

Endenicher Allee 60/0-008 (Mathezentrum)

Given a finite group G, one can count the proportion of pairs of elements in G that commute -- giving a number, denoted dc(G), behaviour of which has been studied since the 1960s.  Such a notion has straightforward generalisations to residually finite or amenable groups.  To make sense of this for other infinite groups G, one needs to define some sort of a nice measure or a mean on G.  In this talk, I will explain how such means -- more specifically, finitely additive probability means that give the "correct" answer for cosets of subgroups -- can be constructed.  As an application of these methods, one can define dc(G) for any group G, and show that dc(G) > 0 if and only if G is finite-by-abelian-by-finite.
This is joint work with Armando Martino.