Commuting probabilities of infinite groups

Abstract

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, 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 m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a sequence of almost-invariant measures on G, we show that if dc_M(G) is at least a>0 then G contains a normal subgroup G' of index f(a) and a normal subgroup H of cardinality at most g(a) such that G'/H is abelian. This generalises a result for finite groups due to P. M. Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolin, Martino and Ventura. We also describe some general conditions on M under which such theorems hold. We close with an application to 'conjugacy ratios' as introduced by Cox.

18 pages. Some theorem, section and reference numbering is significantly changed from previous versions

References in corpus (3)

• Probabilistic nilpotence in infinite groups
• Rational growth and degree of commutativity of graph products
• The conjugacy ratio of groups

Cited by in corpus (7)

• On the commuting probability of p-elements in a finite group
• Probabilistic nilpotence in infinite groups
• Cayley graphs with few automorphisms: the case of infinite groups
• Conjugacy classes of $\pi$-elements and nilpotent/abelian Hall $\pi$-subgroups
• Compact groups with high commuting probability of monothetic subgroups
• Commuting Probability of Finite Groups (Extended)
• The Conjugacy Ratio of Abelian-by-Cyclic Groups