Let $\cal C$ be the smallest class of finitely generated discrete countable groups such that

- $\cal C$ contains all infinite cyclic groups,
- if $G$ is
*any*finitely generated countable discrete group and $H\in \cal C$ then $G\ast H \in \cal C$, - If $G$ is any countable discrete group and $H$ is a finite index subgroup then $G\in {\cal C} \iff H\in \cal C$.

In 1. I write "all infinite cyclic groups" to avoid adding that $\cal C$ is closed under isomorphism, but feel free to replace it with your favorite infinite cyclic group and adding that $\cal C$ is closed under isomorphisms.

Question. Is there a better ("more intrinsic") description of the class $\cal C$?

My motivation: I have been studying when for a given group $G$ is there a finite set $M$ and a subshift of finite type $S\subset M^G$ such that $G$ acts on $S$ freely (not essentially freely, topologically freely, etc, but *freely*). It's not difficult to see groups in class $\cal C$ don't admit free subshifts of finite type, and together with a colleague we can prove that some groups (e.g. polycyclic, but also some with infinitely many ends) not in $\cal C$ admit free subshifts of finite type. It seems natural to conjecture $\cal C$ is the class of groups which don't admit a free subshift of finite type, so I'd like to understand it.