  A group $G$ is said to be intersection-saturated if for every strictly
positive integer $n$ and every map $c\colon \mathcal{P}(\{1,\dots,
n\})\setminus \emptyset \rightarrow \{0,1\}$, one can find subgroups
$H_1,\dots, H_n\leq G$ 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.
