The Σ2-invariants for graph products of indicable groups

Given a finite simplicial graph G and groups Gv for each vertex v∈G, the graph product GG is the free product of the vertex groups modulo the normal closure of all commutator groups [Gv,Gw] where v and w are adjacent vertices. If all vertex groups are indicable we give a complete description of the homotopical invariant Σ2(GG) which captures, among other things, information determinining which normal subgroups above the commutator are finitely presented. In case some vertex group is not indicable, our result still gives an almost complete picture of Σ2(GG).