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

A couple of extra points. Any compact 3manifold with boundary $M$ can be doubled to give a closed 3manifold $D$. 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 BaumslagSolitar groups that Richard mentions above) applies just as well to compact 3manifolds as closed 3manifolds. Other classes of poison subgroups can be constructed from cohomological conditions. The KneserMilnor Theorem implies that any closed, irreducible 3manifold 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 3manifold (and hence of a compact 3manifold, by the previous paragraph). EDIT: Oh, and yet another source of poison subgroups comes from Scott's theorem that 3manifold 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. 


No. The Baumslag solitar groups $\langle a, b  ab^m a^{1} = b^n \rangle$ are not 3manifold groups when $m \neq n$. See Heil, Wolfgang H. Some finitely presented non$3$manifold groups. Proc. Amer. Math. Soc. 53 (1975), no. 2, 497500. (See also Peter Shalen, ThreeManifolds and BaumslagSolitar groups Topology Appl. 110 (2001), 113118) 


I recently heard of a result due to Aitchison and Reeves which shows that any finitely presented group arises as the fundamental group of a 3dimensional orbifold (where fundamental group means the topological and not the orbifold fundamental group). In fact, they say that the orbifold can be taken to be the quotient of a closed oriented hyperbolic 3manifold by an isometric involution with isolated fixed points, all modelled on $x\mapsto x$. (I'm certainly no expert on this topic, just passing on what I heard.) 

