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

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.

v2: Published version taking into account referee suggestions. Slight change in title v1: 11 pages, 2 figures. Comments are welcome