# Random elements of (infinite) finitely generated groups

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.

• 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 $S$.
– YCor
Jan 27, 2021 at 12:54