
> 
> For which $3$-manifolds $M$ is the fundamental group $\pi\_1(M)$
>  finitely generated and has positive rank gradient?
> 
> 
> 


Recall that the rank gradient of a finitely generated group $G$ is defined to be $$\inf\_{H} \frac{d(H) - 1}{[G : H]}$$ where the infimum is taken over all finite index subgroups $H$ of $G$, and $d(H)$ stands for the minimal cardinality of a generating set of $H$.



Which 3-manifolds have positive rank gradient?
This is answered in the proof of Theorem 8.5 of [this paper](https://math.berkeley.edu/sites/default/files/faculty/files/virtualspecialICM.pdf). This
says that the rank gradient is zero iff $M$ is prime or $\mathbb{RP}^3\#\mathbb{RP}^3$. 


**Edit:** There's a small step missing from the argument. The argument shows that if $G$ is an orientable connected sum 3-manifold group (not $\mathbb{RP}^3\#\mathbb{RP}^3$), then there exists $G' \lhd G$ of finite index so that $G'$ has positive corank gradient, hence positive rank gradient. Since $d(G')-1\leq [G:G'](d(G)-1)$, if $G$ has rank gradient $0$, so would $G'$ a contradiction (one applies this inequality to $G'\cap H \lhd H$). This applies as well to non-orientable 3-manifolds, where one has to consider the $\mathbb{RP}^2$-decomposition, or just posit that there is a prime cover. 


