On groups whose cogrowth series is the diagonal of a rational series

TL;DR

This paper proves that groups with a finite-index subgroup isomorphic to ^n imes F_m have cogrowth series that are diagonals of rational functions, extending known results to a broader class of groups.

Contribution

It establishes that such groups' cogrowth series are diagonals of rational functions, answering a question about virtually abelian groups and generalizing previous results.

Findings

Cogrowth series of these groups are diagonals of rational functions.

Answers a question on virtually abelian groups' cogrowth series.

Generalizes results to Baumslag-Solitar groups (N,N).

Abstract

We show that if a group contains $\mathbb{Z}^n \times F_m$ as a finite-index subgroup, then its cogrowth series is the diagonal of a rational function for every generating set. This answers a question of Pak and Soukup on the cogrowth of virtually abelian groups; and generalises a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag-Solitar groups $\mathrm{BS}(N,N)$.