Let $G$ be a finitely generated residually finite group with positive rank gradient, and let $F_2$ be the free group on $2$ elements. Must there be an embedding $i \colon F_2 \to G$ ?

A group $G$ is called residually finite if the intersection of all of its finite index subgroups is trivial.

A group $G$ is said to have positive rank gradient if the rank of finite index subgroups grows linearly with the index. More formally, the Rank Gradient (RG) of a finitely generated group $G$ is defined to be:

$$\mathrm{RG}(G) = \inf_{H} \frac{\mathrm{rank}(H) - 1}{[G : H]}$$ where $H$ runs over all subgroups of finite index in $G$, and the rank of a group is the smallest cardinality of a generating set for the group.

The notion of rank gradient is also important in the theory of $3$-manifolds.