TITLE: Is there a nonabelian free group inside a group of positive rank gradient?
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: Lackenby ( http://people.maths.ox.ac.uk/lackenby/lg070105.ps ) proved that a finitely presented group, which has a pro-$p$ completion of positive rank gradient, is large, i.e. it contains a subgroup of finite index that projects onto a non-abelian free group. So while the answer to the original question is no, if you add some extra conditions it becomes yes in a very strong sense.