Deficiency and abelianized deficiency of some virtually free groups

Abstract

Let Q m 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 Γ m,n = Q m * Q n. We show that for certain choices of (m, n), for example (2, 3), the group Γ 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 Γ m,n has a relation gap. © 2007 Cambridge Philosophical Society.