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

We show that if a group contains ℤ n × 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 generalizes a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag–Solitar groups BS (N , N). [ABSTRACT FROM AUTHOR]