Degree of commutativity of infinite groups

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 28 Jul 2016Embargo end date: 01 Jan 2015 Spain, United Kingdom English Publisher:American Mathematical Society (AMS)Journal:Proceedings of the American Mathematical Society, volume 145, pages 479-485 (issn: 0002-9939, eissn: 1088-6826,

Copyright policy )Funded by:EC | PCG

Authors: Antolin, Yago; Martino, Armando; Ventura Capell, Enric;

doi: 10.1090/proc/13231 , 10.48550/arxiv.1511.07269

arXiv: 1511.07269

handle: 2117/101873

Degree of commutativity of infinite groups

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Abstract

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.

11 pages

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Spain, United Kingdom

Related Organizations

University of Southampton
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Keywords

polynomial growth, abelian groups, Grups abelians, Group Theory (math.GR), 510, Grups infinits, Probabilistic methods in group theory, FOS: Mathematics, Classificació AMS::20 Group theory and generalizations::20P05 Probabilistic methods in group theory, :20 Group theory and generalizations::20P05 Probabilistic methods in group theory [Classificació AMS], Residual properties and generalizations; residually finite groups, Generators, relations, and presentations of groups, Teoria de, Polynomial growth, Grups, Grups, Teoria de, Degree of commutativity, :Matemàtiques i estadística::Àlgebra [Àrees temàtiques de la UPC], 004, Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra, Infinite groups, degree of commutativity, Group theory, Mathematics - Group Theory, Abelian groups

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