Title: The homotopy type of a finite 2-complex with non-minimal Euler characteristic

Abstract: Two presentations for a group G which have the same deficiency are called exotic if the corresponding presentation complexes are not homotopy equivalent. The first examples of exotic presentations were found by Dunwoody and Metzler in the 1970s but, owing to the difficulty of the algebra involved, few other examples have since been found. We show that, for each k, there exists exotic presentations for which the corresponding presentation complexes have Euler characteristic k above the minimal value. This resolves Problem D5 in the 1979 Problems List of C. T. C. Wall.

We also establish the closely related fact that, for all k ≥ 1, there exists a group G and a non-free stably free ZG-module of rank k. The basis for our proof is a remark made in a 1999 paper by Martin Evans.