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, , H₍ G such that for every non-empty subset I \1, , n\, the intersection ₈ ₈H₈ 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.