Using stably free non-free relation modules we construct an infinite collection of 2-dimensional homotopy types, each of Euler-characteristic one and with trefoil fundamental group. This provides an affirmative answer to a question asked by Berridge and Dunwoody [1]. We also give new examples of exotic relation modules. We show that the relation module associated with the generating set {x, y 4 } for the Baumslag-Solitar group x, y | xy 2 x −1 = y 3 is stably free non-free of rank one. 57M20; 57M05

A labeled oriented tree is called injective, if each vertex occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof uses a new relative asphericity test based on a lemma of Stallings.MSC: 57M20, 57M05, 20F05

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The generation gap of a group G is the difference between the minimal number of generators of G and the rank of the augmentation ideal. The relation gap of a presentation FrN is the difference between the minimal number of elements that generate N as a normal subgroup and the minimal number of G-module generaw x tors of the relation module Nr N, N . We show that if G is a finitely presented group then there exists n such that G = Ł n Z , Z being the cyclic group of is 1 p p order p, has zero generation and zero relation gap. We apply this result to questions concerning the efficiency of finite groups. ᮊ