I have been studying connections between Geometric Group Theory and Probability, and I was wondering: what work is there about random elements of finitely generated groups?

More formally, let $G=\langle S,R\rangle$ be a finitely generated group, and let $d$ be the metric on $G$ associated with that presentation. Let $p_n$ be a uniform distribution on all elements $x$ with $d(x,e)\leq n.$ What can we say about the probabilities of properties (for example, being contained within a certain fixed subgroup) of elements chosen this way? Furthermore, what about the asymptotes of these probabilities as $n\to\infty?$

If anyone can give references to papers/books discussing these sorts of questions, I would greatly appreciate it.

  • 1
    $\begingroup$ One of the most active probabilistic study of groups is the study of random walks on groups. A drawback of your approach using uniform probability on balls is that such questions can be sensitive on the choice of $S$. $\endgroup$
    – YCor
    Jan 27, 2021 at 12:54

1 Answer 1


Consider Haar measure on locally compact groups, including profinite groups.

Your proposed measure is studied in these references.
Borovik, Alexandre V. and Myasnikov, Alexei G. and Shpilrain, Vladimir, Measuring sets in infinite groups, 2002.

Alexandre V. Borovik, Alexei G. Myasnikov, Vladimir N. Remeslennikov, Multiplicative measures on free groups, 2002.


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