Algebraic and arithmetic properties of the cogrowth sequence of nilpotent groups

We prove that congruences of the cogrowth sequence in a unitriangular group UT$(m,\mathbb{Z})$ are undecidable. This is in contrast with abelian groups, where the congruences of the cogrowth sequence are decidable. As an application, we conclude that there is no algorithm to present the cogrowth series as the diagonal of a rational function.