Deficiency and abelianized deficiency of some virtually free groups

Abstract

Let $Q_m$ be the HNN extension of $\mathbb{Z}/m\times \mathbb{Z}/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups ${\Gamma }_{m,n}=Q_m\ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group ${\Gamma }_{m,n}$ has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if ${\Gamma }_{m,n}$ has a relation gap.

7 pages; no figures. Minor changes; now to appear in Math. Proc. Camb. Phil. Soc

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