Home / Events / Talk: "The degree of commutativity of an in finite group" (E. Ventura, UPC)

Talk: "The degree of commutativity of an in finite group" (E. Ventura, UPC)

Abstract. There is a classical result saying that, in a nite 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 in nite groups, but it is possible to give a notion of degree of commutativity for nitely generated groups (w.r.t. a xed nite 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 nite setting, but we are still able to prove some results concerning this new concept, the main one being the following: for any nitely generated group of polynomial growth G, the commuting degree of G is positive if and only if G is virtually abelian.

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