Generation Gaps and Abelianised Defects of Free Products

Abstract

Let G be a group of the form G_1* ... *G_n, the free product of n subgroups, and let M be a ZG-module of the form $\bigoplus_{i=1}^n M_i \otimes_{\mathbb{Z}G_i} \mathbb{Z}G$. We shall give formulae in various situations for $d_{ZG}(M)$, the minimum number of elements required to generate M. In particular if C_1,C_2 are non-trivial finite cyclic groups of coprime orders, $G = (C_1 \times Z) * (C_2 \times Z)$ and $F/R \cong G$ is the free presentation obtained from the natural free presentations of the two factors, then the number of generators of the relation module, $d_{\mathbb{Z}G}(R/R')$ is three. It seems plausible that the minimum number of relators of G should be 4, and this would give a finitely presented group with positive relation gap. However we cannot prove this last statement.