A virtually nilpotent group whose Green series is not D-finite

TL;DR

This paper presents the first example of a virtually nilpotent group with a specific generating set whose Green series is not D-finite, using arithmetical properties and subword complexity analysis.

Contribution

It introduces a novel example of a virtually nilpotent group with a non-D-finite Green series, expanding understanding of group growth series.

Findings

The Green series of the constructed group is not D-finite.

The proof involves an arithmetical miracle and subword complexity analysis.

This is the first such example in virtually nilpotent groups.

Abstract

We provide the first example of virtually nilpotent group, with a specific generating set, for which the Green series (sometimes called cogrowth series) is not $D$-finite. The proof relies on an arithmetical miracle, and the study of the subword complexity of a multiplicative sequence coming out of it.