Soluble Lie Rings

In this chapter, we present some calculations of zeta functions of soluble (but non-nilpotent) Lie rings over ℤ. Since these Lie rings are not nilpotent, the Mal'cev correspondence cannot be used, and so there is no corresponding ℑ-group whose local zeta functions we are also calculating. We prove that the zeta functions we consider behave in a similar fashion to those of nilpotent Lie rings. What is remarkable is that the uniform behaviour is ‘stronger’ than that seen with the nilpotent Lie rings.