Algebraic and arithmetic properties of the cogrowth sequence of nilpotent groups

TL;DR

This paper investigates the algebraic and arithmetic properties of the cogrowth sequence in nilpotent groups, proving undecidability of certain congruences and contrasting with decidable cases in abelian groups.

Contribution

It establishes the undecidability of congruences of the cogrowth sequence in unitriangular groups, highlighting fundamental differences from abelian groups.

Findings

Congruences of the cogrowth sequence in UT(m, Z) are undecidable.

Contrasts with decidable cases in abelian groups.

No algorithm exists to represent the cogrowth series as a diagonal of a rational function.

Abstract

We prove that congruences of the cogrowth sequence in a unitriangular group UT$(m, \Bbb Z)$ 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.