Intersection-saturated groups without free subgroups

Abstract

A group $G$ is said to be intersection-saturated if for every strictly positive integer $n$ and every map $c:\mathcal{P}(\{1,\dots ,n\})\setminus \varnothing \to \{0,1\}$, one can find subgroups $H_1,\dots ,H_n\le G$ such that for every non-empty subset $I\subseteq \{1,\dots ,n\}$, the intersection ${\bigcap }_{i\in I}{H}_i$ 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.