Coset correct means on groups and the probability that two elements commute
Armando Martino, Motiejus Valiunas
TL;DR
This work introduces coset correct means (CCMs), a relaxed form of invariant means that are left invariant on subgroups and their cosets, and proves that every group admits CCMs via two distinct constructions. Using CCMs, it defines the degree of commutativity dc_ (G) as the CCM-based probability that two elements commute, showing that positive dc_ (G) occurs if and only if the group is finite-by-abelian-by-finite (FAF), with the value being independent of the CCM for FAF groups. The paper then links these notions to familiar quotient and profinite structures: for residually finite G, dc_RF(G) > 0 iff G is virtually abelian, and dc_RF(G) equals the CCM-based value via Haar measure on the profinite completion; in amenable groups, positivity of the degree also characterizes FAF, while a defect δ_G is shown to be a rigid 0/1 invariant reflecting amenability. Finally, the authors explore extensions to conjugacy-class representatives and present a counterexample showing the limits of product versus non-product CCMs in determining degree-of-commutativity in non-amenable settings. Overall, the paper unifies previous degree-of-commutativity results across finite, residually finite, and amenable groups through the CCM framework, and reveals a deep connection between coset-invariant means and group structure.
Abstract
Amenable groups are those admitting an invariant mean -- a finitely additive probability mean that assigns equal ``weight'' to any two translates of the same set. We introduce coset correct means (CCMs), a class of finitely additive means that, for any subgroup, assigns equal weight to all its cosets, weakening and therefore generalising the notion of an invariant mean. We show that, unlike the case for invariant means, every group admits a CCM and give two constructions -- one via the Ultrafilter Lemma and one via the Hahn--Banach Theorem -- both relying on a Theorem of B. H. Neumann. Using CCMs, we define a degree of commutativity for arbitrary groups, measuring the ``probability'' that two random elements of a group commute. We prove that this degree of commutativity is independent of the choice of CCM and is positive precisely for finite-by-abelian-by-finite groups, recovering and unifying previous characterisations. We also introduce a defect function that quantifies the failure of left invariance for finitely additive means, and define the defect of a group as the infimum of these. We then prove a dichotomy: the defect for a group is either 0 or 1, with 0 characterising amenable groups.