Is it true that *any finitely presented group can be realized as fundamental group of compact 3-manifold **with boundary***?



Fundamental group of 3-manifold with boundary
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 $\pi\_1(D)$. Therefore, any "poison subgroup" (such as the Baumslag--Solitar groups that Autumn mentions above) applies just as well to compact 3-manifolds as closed 3-manifolds.


Other classes of poison subgroups can be constructed from cohomological conditions. The Kneser--Milnor Theorem implies that any closed, irreducible 3-manifold with infinite fundamental group is aspherical. It follows that any freely indecomposable infinite group with cohomologial dimension greater than 3 cannot be a subgroup of a closed 3-manifold (and hence of a compact 3-manifold, by the previous paragraph).


EDIT:


Oh, and yet *another* source of poison subgroups comes from Scott's theorem that 3-manifold groups are *coherent*, meaning that every finitely generated subgroup is finitely presented. This rules out subgroups like $F\times F$ (where $F$ is a free group), which is not coherent.


