Year: 2020
Type: article
Abstract: Let G be a group, and let M = ( μ n ) n = 1 ∞ be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to μ n commute. Antolín, Martino and Ventura define the degree of commutativity dc M ( G ) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if μ n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and ( μ n ) is a sequence of almost-invariant measures on G, we show that if dc M ( G ) ⩾ α > 0 , then G contains a normal subgroup Γ of index at most ⌈ α − 1 ⌉ and a normal subgroup H of cardinality at most ( 2 α − 1 ) O ( α − 2 log α − 1 ) such that H ⊂ Γ and such that Γ / H is abelian. This generalises a result for finite groups due to Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolín, Martino and Ventura. We also describe some general conditions on ( μ n ) under which such theorems hold. We close with an application to conjugacy ratios as introduced by Cox.
DOI: 10.1112/jlms.12305
Author: Matthew C. H. Tointon
Institution: University of Cambridge
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2763561309