I believe the following statement is true, and I've even seen it referenced [here](https://mathoverflow.net/questions/31904/closed-3-manifolds-with-free-abelian-fundamental-groups). Could someone point me to a proof?



> 
> The fundamental group of a closed hyperbolic 3-manifold is not a free product.
> 
> 
> 



Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable
If $M$ is a closed $3$-manifold and $\pi\_1(M) \cong A \ast B$ with $A$ and $B$ nontrivial, then Kneser's conjecture (which is a theorem -- the proof can be found in Hempel's book on 3-manifolds) says that we can write $M = M' \sharp M''$ where $M'$ and $M''$ are closed 3-manifolds with $\pi\_1(M')=A$ and $\pi\_1(M'')=B$. In particular, $M$ contains an embedded $2$-sphere which does not bound a ball (namely, the sphere from the connect sum decomposition). However, all embedded $2$-spheres in hyperbolic 3-manifolds bound balls, as can be seen by lifting to the universal cover.


