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[17/4/17: edited to correct proof.] This is true. First, we need a lemma which builds a related cover. Throughout, $\alpha$ is a simple closed geodesic of length $\ell$, and $\beta_1,\ldots,\beta_n$ …

This question is certainly open in general. I don't know if anyone has formally expressed an 'expectation' in print, but you might be interested in the following pieces of positive evidence. I believ …

The free subgroup generated by $a_1,b_1$ is carried by a one-holed torus $T$ embedded in the surface. Let $\gamma$ be a simple closed curve embedded in $T$, and consider $T\smallsetminus \gamma$. A …

This question is very vague, but here are some thoughts to add to Mark's answer. First, note that any finitely presented group arises as the fundamental group of a closed manifold of dimension 4 (see …

One can also see it using the theory of ends. If $\pi_1M$ were freely decomposable, then it would follow from the easy direction of Stallings' Ends Theorem that $\pi_1M$ had two or infinitely many en …

A couple of extra points. Any compact 3-manifold with boundary $M$ can be doubled to give a closed 3-manifold $D$. As $M$ is a retract of $D$, it follows that $\pi_1(M)$ injects into $\pi_1(D)$. The …