Intersection-saturated groups without free subgroups

A group G is said to be intersection-saturated if for every strictly positive integer n and every map c P(\{1,…, n\}) ∅ → \{0,1\} , one can find subgroups H₁,…, Hₙ≤ G such that for every non-empty subset I⊆ \{1,…, n\} , the intersection i∈ IHᵢ is finitely generated if and only if c(I)=0 . We obtain a new criterion for a group to be intersection-saturated based on the existence of arbitrarily high direct powers of a subgroup admitting an automorphism with a non-finitely generated set of fixed points. We use this criterion to find new examples of intersection-saturated groups, including Thompson’s groups and the Grigorchuk group. In particular, this proves the existence of finitely presented intersection-saturated groups without non-abelian free subgroups, thus answering a question of Delgado, Roy and Ventura.