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H 1 The degree of commutativity of an infinite group

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

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    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, Pascal
    Dates : 14/09/15 - 18/09/15
    Année de la rencontre : 2015
    URL Congrès : http://conferences.cirm-math.fr/1212.html

    Citation 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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