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Can a Poincaré duality group $G$ contain Baumslag--Solitar subgroups $H$ such as BS(1,3) or BS(2,3)?

I don't mean to include those subgroups which are the fundamental group of the torus or Klein bottle. Also, one can show that index must be infinite: $[G:H] = \infty$.

If so, what is a simple example of such $G$ and $H$?

In any case, please let me know of references. Thanks!

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1 Answer 1

up vote 8 down vote accepted

The presentation complex of $BS(m,n)$ is aspherical (as is the presentation complex of any torsion-free 1-related group, see Lyndon and Schupp, "Combinatorial group theory", page 161). Hence by the Davis construction, $BS(m,n)$ embeds into the fundamental group of a closed aspherical 4-dim. manifold which is a PD group . An explicit presentation of that bigger group may be not easy.

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