
> 
> 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.



Is there a nonabelian free group inside a group of positive rank gradient?
In 


Denis Osin, Rank gradient and torsion groups. 
Bull. Lond. Math. Soc. 43 (2011), no. 1, 10–16,


the following theorem is proved



> 
> **Theorem** For every prime $p$, there exists a finitely generated infinite residually finite $p$-group with positive rank gradient.
> 
> 
> 


<http://arxiv.org/abs/0905.1322>


<http://blms.oxfordjournals.org/content/43/1/10>


