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H 1 The degree of commutativity of an infinite group

Auteurs : Ventura, Enric (Auteur de la Conférence)
CIRM (Editeur )

Résumé : There is a classical result saying that, in a finite group, the probability that two elements commute is never between $5/8$ and 1 (i.e., if it is bigger than $5/8$ then the group is abelian). It seems clear that this fact cannot be translated/adapted to infinite groups, but it is possible to give a notion of degree of commutativity for finitely generated groups (w.r.t. a fixed finite set of generators) as the limit of such probabilities, when counted over successively growing balls in the group. This asymptotic notion is a lot more vague than in the finite setting, but we are still able to prove some results concerning this new concept, the main one being the following: for any finitely generated group of polynomial growth $G$, the commuting degree of $G$ is positive if and only if $G$ is virtually abelian.

Codes MSC :
20P05 - Probabilistic methods in group theory

 Informations sur la Vidéo Langue : Anglais Date de publication : 13/10/15 Date de captation : 17/09/15 Collection : Research talks ; Algebra Format : MP4 ; audio/x-aac Durée : 00:49:22 Domaine : Algebra Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-09-17_Ventura.mp4 Informations sur la rencontre Nom de la rencontre : GAGTA-9: geometric, asymptotic and combinatorial group theory and applications / GAGTA-9 : Théorie géométrique, asymptotique et combinatoire des groupes et applications Organisateurs de la rencontre : Coulbois, Thierry ; Weil, PascalDates : 14/09/15 - 18/09/15 Année de la rencontre : 2015 URL Congrès : http://conferences.cirm-math.fr/1212.htmlCitation Data DOI : 10.24350/CIRM.V.18838503 Cite this video as: Ventura, Enric (2015). The degree of commutativity of an infinite group. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18838503 URI : http://dx.doi.org/10.24350/CIRM.V.18838503

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