Geometric realisation of relation modules over aspherical groups

Abstract

We prove that the direct sums of extensions of scalars of relation modules are geometrically realisable as the second homotopy group of a finite 2-complex. We use this to exhibit a finite 2-complex with fundamental group the (10,15) torus knot group and non-free π2, yielding exotic presentations of a group for which no such examples had previously been known. We conclude by constructing stably free non-free modules over an infinite family of Baumslag-Solitar groups; it remains to determine whether these modules are geometrically realisable by finite 2-complexes.