Lior Bary-Soroker: The Diamond Theorem for Finitely Generated Free Profinite Groups Abstract: In this talk we discuss when a closed subgroup of a free profinite group is free. We introduce the notion of a sparse subgroup and prove that if $M$ is a sparse subgroup of a free group of finite or countable rank, then $M$ is free. The main result of the talk is the so-called Haran's Diamond Theorem for finitely generated free profinite groups.