For a diagram group G, the first derived quotient G1/G2 is always free abelian (as proved by M. Sapir and V. Guba). However the second derived quotient G2/G3 may contain torsion. In fact, we show that for any finite or countably infinite direct product of cyclic groups A, there is a diagram group with second derived quotient A. We use that to construct families with the properties of the title.
Cite this article
Stephen J. Pride, Universal diagram groups with identical Poincaré series. Groups Geom. Dyn. 4 (2010), no. 4, pp. 901–908DOI 10.4171/GGD/113