Commuting probabilities of infinite groups
| Autor: | Matthew Tointon |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Normal subgroup
Sequence Degree (graph theory) Distribution (number theory) Group (mathematics) General Mathematics Probability (math.PR) 010102 general mathematics Group Theory (math.GR) 01 natural sciences Combinatorics Mathematics::Group Theory Conjugacy class 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Abelian group Mathematics - Group Theory Mathematics - Probability Mathematics Probability measure |
| Zdroj: | Journal of the London Mathematical Society. 101:1280-1297 |
| ISSN: | 1469-7750 0024-6107 |
| Popis: | 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 |
| Databáze: | OpenAIRE |
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