Studying actions of groups on probability spaces leads to many questions that are of serious interest to group-theorists and can be understood and contributed to by probabilists. In particular, the theory of cost has raised several outstanding open questions that can be interpreted as questions about uniform spanning forests and factor of IID processes. I will introduce these problems and give a solution to one of them, namely that Kazhdan groups have cost 1. All the technical terms in the previous sentence will be introduced in a way accessible to probabilists. Joint work with Gabor Pete.