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SUMMARY:What is the probability that two elements of a group commute? - Ar
 mando Martino (Southampton)
DTSTART:20260227T134500Z
DTEND:20260227T144500Z
UID:TALK243316@talks.cam.ac.uk
CONTACT:Francesco Fournier-Facio
DESCRIPTION:This is mostly joint work with Motiejus Valiunas but concerns 
 a range of questions that I have thought about for a little time and will 
 also mention the work of Yago Antolin\, Matthew Tointon and Enric Ventura.
 \n\nThis will be a recap of various results and possible methods for assig
 ning a probability to a group\, measuring the event that two elements comm
 ute. I will quickly recap a classical result from finite groups\, then mov
 e on to residually finite groups and amenable groups before outlining a ne
 w method for calculating this probability via what we call "coset correct 
 means". A coset correct mean is a function that assigns to each subset of 
 a group a number between 0 and 1\, which is finitely additive\, assigns 1 
 to the whole group and gives the "correct" answer for any coset of any sub
 group (that number being the reciprocal of the index of the subgroup). I w
 ill outline the construction of these coset correct means.\n\nThe main res
 ult in these contexts is that the probability that two elements of a group
  commute is non-zero exactly when the group itself is FAF - finite-by-abel
 ian-by-finite. In many situations - such as when the group is finitely gen
 erated - a group is FAF just means that it is virtually abelian. That is a
  finitely generated FAF group is one that has an abelian subgroup of finit
 e index.
LOCATION:MR13
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