Is it true that any finitely presented group can be realized as fundamental group of compact 3-manifold with boundary?
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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 Richard 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. |
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No. The Baumslag solitar groups $\langle a, b | ab^m a^{-1} = b^n \rangle$ are not 3-manifold groups when $m \neq n$. See Heil, Wolfgang H. Some finitely presented non-$3$-manifold groups. Proc. Amer. Math. Soc. 53 (1975), no. 2, 497--500. (See also Peter Shalen, Three-Manifolds and Baumslag-Solitar groups Topology Appl. 110 (2001), 113--118) |
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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 3-dimensional 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 3-manifold 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.) |
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