Infinite branching in the first syzygy

The first syzygy Ω1(Z) of a group G consists of the isomorphism classes of modules which are stably equivalent to the augmentation ideal I=Ker(ϵ:Z[G]→Z). When G is finitely generated Ω1(Z) admits the structure of an infinite tree whose roots do not extend infinitely downward. We show that the minimal level is infinite for certain groups of the form where Φ is finite.