The Schur multiplier of a generalized Baumslag-Solitar group. (Q641050)

The Baumslag-Solitar groups \(BS(m,n)=\langle t,x\mid (x^m)^t=x^n\rangle\) arise as a graph of groups with infinite cyclic edge and vertex groups. Generalizing these, a generalized Baumslag-Solitar group (GBS-group) is the fundamental group of any (finite, connected) graph of groups with infinite cyclic edge and vertex groups. The main theorem of this paper is that the Schur multiplier of a GBS-group \(G\) is free Abelian of rank \(r_0(G)-1\) where \(r_0(G)\) is the torsion-free rank of \(G_{\mathrm{ab}}\). A second theorem gives the value of \(r_0(G)\). Corollaries give that every GBS-group has Euler characteristic 0 and deficiency 1, and give necessary and sufficient conditions for the Schur multiplier to be trivial. As an application it is determined when all central extensions of an Abelian group by a GBS-group are direct products.