Universal diagram groups with identical Poincaré series

Abstract

Abstract. For a diagram group G, the first derived quotient G 1 =G 2 is always free abelian (as proved by M. Sapir and V. Guba). However the second derived quotient G 2 =G 3 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.