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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.

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    $\begingroup$ 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

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Consider Haar measure on locally compact groups, including profinite groups.
https://en.wikipedia.org/wiki/Haar_measure#A_construction_using_compact_subsets
https://mathoverflow.net/questions/97971/haar-measure-for-profinite-groups-reference-needed

Your proposed measure is studied in these references.
Borovik, Alexandre V. and Myasnikov, Alexei G. and Shpilrain, Vladimir, Measuring sets in infinite groups, 2002.
https://arxiv.org/abs/math/0204078

Alexandre V. Borovik, Alexei G. Myasnikov, Vladimir N. Remeslennikov, Multiplicative measures on free groups, 2002.
https://arxiv.org/abs/math/0204070

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