An interesting fact was relayed to me in another question of mine that

If $M$ is any closed manifold with universal cover homeomorphic to $R^n$ for $n>1$ then $\pi_1(M)$ is freely indecomposable.

What are some other sufficient conditions for the free-indecomposability of a group? Are there any interesting necessary conditions?

An interesting fact was relayed to me in another question of mine that

If $M$ is any closed manifold with universal cover homeomorphic to $R^n$ for $n>1$ then $\pi_1(M)$ is freely indecomposable.

What are some other sufficient conditions for the free-indecomposability of a group? Are there any interesting necessary conditions?