Deficiency and abelianized deficiency of some virtually free groups

Abstract

AbstractLet Qm be the HNN extension of Z/m × Z/m where the stable letter conjugates the first factor to the second. We explore small presentations of the groups Γm,n = Qm* Qn. We show that for certain choices of (m,n), for example (2,3), the group Γm,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 Γm,n has a relation gap.