Generation gaps and abelianized defects of free products
K. W. Gruenberg, Peter A. Linnell · Journal of Group Theory · 2008
Let G be a group of the form , the free product of n subgroups, and let M be a ℤG -module of the form . 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, and F/R ≅ 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 ℤG ( R/R ′), is 3. 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.